id	sid	tid	token	lemma	pos
cana-671	1	1	communications	communication	NOUN
cana-671	1	2	on	on	ADP
cana-671	1	3	applied	apply	VERB
cana-671	1	4	nonlinear	nonlinear	ADJ
cana-671	1	5	analysis	analysis	NOUN
cana-671	1	6	issn	issn	NOUN
cana-671	1	7	:	:	PUNCT
cana-671	1	8	1074	1074	NUM
cana-671	1	9	-	-	PUNCT
cana-671	1	10	133x	133x	NUM
cana-671	1	11	vol	vol	NOUN
cana-671	1	12	31	31	NUM
cana-671	1	13	no	no	NOUN
cana-671	1	14	.	.	PUNCT
cana-671	2	1	2s	2s	NUM
cana-671	2	2	(	(	PUNCT
cana-671	2	3	2024	2024	NUM
cana-671	2	4	)	)	PUNCT
cana-671	2	5	572	572	NUM
cana-671	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-671	2	7	physics	physics	NOUN
cana-671	2	8	and	and	CCONJ
cana-671	2	9	mechanics	mechanic	NOUN
cana-671	2	10	use	use	VERB
cana-671	2	11	geometric	geometric	ADJ
cana-671	2	12	algebra	algebra	NOUN
cana-671	2	13	minavere	minavere	PROPN
cana-671	2	14	hasani	hasani	PROPN
cana-671	2	15	berisha1	berisha1	PROPN
cana-671	2	16	,	,	PUNCT
cana-671	2	17	rushadije	rushadije	PROPN
cana-671	2	18	ramani	ramani	PROPN
cana-671	2	19	halili2	halili2	PROPN
cana-671	3	1	1assistant	1assistant	NUM
cana-671	3	2	department	department	NOUN
cana-671	3	3	of	of	ADP
cana-671	3	4	faculty	faculty	NOUN
cana-671	3	5	of	of	ADP
cana-671	3	6	mathematical	mathematical	ADJ
cana-671	3	7	and	and	CCONJ
cana-671	3	8	natural	natural	ADJ
cana-671	3	9	sciences	sciences	PROPN
cana-671	3	10	university	university	PROPN
cana-671	3	11	of	of	ADP
cana-671	3	12	tetova	tetova	NUM
cana-671	3	13	ilinden	ilinden	ADJ
cana-671	3	14	street	street	NOUN
cana-671	3	15	no.1200	no.1200	PROPN
cana-671	3	16	tetovo	tetovo	PROPN
cana-671	3	17	,	,	PUNCT
cana-671	3	18	republic	republic	NOUN
cana-671	3	19	of	of	ADP
cana-671	3	20	north	north	PROPN
cana-671	3	21	macedonia	macedonia	PROPN
cana-671	3	22	,	,	PUNCT
cana-671	3	23	email	email	NOUN
cana-671	3	24	:	:	PUNCT
cana-671	3	25	m.hasani-berisha211497@unite.edu.mk	m.hasani-berisha211497@unite.edu.mk	PROPN
cana-671	3	26	(	(	PUNCT
cana-671	3	27	author	author	NOUN
cana-671	3	28	)	)	PUNCT
cana-671	3	29	orcid	orcid	NOUN
cana-671	3	30	:	:	PUNCT
cana-671	3	31	0009	0009	NUM
cana-671	3	32	-	-	SYM
cana-671	3	33	0009	0009	NUM
cana-671	3	34	-	-	PUNCT
cana-671	3	35	17159046	17159046	NUM
cana-671	3	36	2professor	2professor	NUM
cana-671	3	37	department	department	NOUN
cana-671	3	38	of	of	ADP
cana-671	3	39	faculty	faculty	NOUN
cana-671	3	40	of	of	ADP
cana-671	3	41	mathematical	mathematical	ADJ
cana-671	3	42	and	and	CCONJ
cana-671	3	43	natural	natural	ADJ
cana-671	3	44	sciences	sciences	PROPN
cana-671	3	45	university	university	PROPN
cana-671	3	46	of	of	ADP
cana-671	3	47	tetova	tetova	NUM
cana-671	3	48	ilinden	ilinden	ADJ
cana-671	3	49	street	street	NOUN
cana-671	3	50	no.1200	no.1200	PROPN
cana-671	3	51	tetovo	tetovo	PROPN
cana-671	3	52	,	,	PUNCT
cana-671	3	53	republic	republic	NOUN
cana-671	3	54	of	of	ADP
cana-671	3	55	north	north	PROPN
cana-671	3	56	macedonia	macedonia	PROPN
cana-671	3	57	,	,	PUNCT
cana-671	3	58	email	email	NOUN
cana-671	3	59	:	:	PUNCT
cana-671	3	60	rushadije.ramani@unite.edu.mk(correspondent	rushadije.ramani@unite.edu.mk(correspondent	NOUN
cana-671	3	61	author).orcid	author).orcid	NOUN
cana-671	3	62	:	:	PUNCT
cana-671	3	63	00000001	00000001	NUM
cana-671	3	64	-	-	SYM
cana-671	3	65	5397	5397	NUM
cana-671	3	66	-	-	SYM
cana-671	3	67	1896	1896	NUM
cana-671	3	68	article	article	NOUN
cana-671	3	69	history	history	NOUN
cana-671	3	70	:	:	PUNCT
cana-671	3	71	received	receive	VERB
cana-671	3	72	:	:	PUNCT
cana-671	3	73	18	18	NUM
cana-671	3	74	-	-	SYM
cana-671	3	75	03	03	NUM
cana-671	3	76	-	-	PUNCT
cana-671	3	77	2024	2024	NUM
cana-671	3	78	revised	revise	VERB
cana-671	3	79	:	:	PUNCT
cana-671	3	80	10	10	NUM
cana-671	3	81	-	-	SYM
cana-671	3	82	05	05	NUM
cana-671	3	83	-	-	PUNCT
cana-671	3	84	2024	2024	NUM
cana-671	3	85	accepted	accept	VERB
cana-671	3	86	:	:	PUNCT
cana-671	3	87	26	26	NUM
cana-671	3	88	-	-	SYM
cana-671	3	89	05	05	NUM
cana-671	3	90	-	-	PUNCT
cana-671	3	91	2024	2024	NUM
cana-671	3	92	abstract	abstract	NOUN
cana-671	3	93	:	:	PUNCT
cana-671	3	94	this	this	DET
cana-671	3	95	paper	paper	NOUN
cana-671	3	96	offers	offer	VERB
cana-671	3	97	a	a	DET
cana-671	3	98	comprehensive	comprehensive	ADJ
cana-671	3	99	overview	overview	NOUN
cana-671	3	100	of	of	ADP
cana-671	3	101	geometric	geometric	ADJ
cana-671	3	102	algebra	algebra	NOUN
cana-671	3	103	.	.	PUNCT
cana-671	4	1	the	the	DET
cana-671	4	2	goal	goal	NOUN
cana-671	4	3	is	be	AUX
cana-671	4	4	to	to	PART
cana-671	4	5	impart	impart	VERB
cana-671	4	6	comprehension	comprehension	NOUN
cana-671	4	7	of	of	ADP
cana-671	4	8	fundamental	fundamental	ADJ
cana-671	4	9	principles	principle	NOUN
cana-671	4	10	and	and	CCONJ
cana-671	4	11	the	the	DET
cana-671	4	12	practical	practical	ADJ
cana-671	4	13	application	application	NOUN
cana-671	4	14	of	of	ADP
cana-671	4	15	geometric	geometric	ADJ
cana-671	4	16	algebra	algebra	NOUN
cana-671	4	17	in	in	ADP
cana-671	4	18	physics	physics	NOUN
cana-671	4	19	and	and	CCONJ
cana-671	4	20	mechanics	mechanic	NOUN
cana-671	4	21	.	.	PUNCT
cana-671	5	1	geometric	geometric	ADJ
cana-671	5	2	algebra	algebra	NOUN
cana-671	5	3	is	be	AUX
cana-671	5	4	a	a	DET
cana-671	5	5	mathematical	mathematical	ADJ
cana-671	5	6	discipline	discipline	NOUN
cana-671	5	7	that	that	PRON
cana-671	5	8	explores	explore	VERB
cana-671	5	9	the	the	DET
cana-671	5	10	interplay	interplay	NOUN
cana-671	5	11	between	between	ADP
cana-671	5	12	algebraic	algebraic	ADJ
cana-671	5	13	concepts	concept	NOUN
cana-671	5	14	and	and	CCONJ
cana-671	5	15	geometric	geometric	ADJ
cana-671	5	16	principles	principle	NOUN
cana-671	5	17	.	.	PUNCT
cana-671	6	1	the	the	DET
cana-671	6	2	fields	field	NOUN
cana-671	6	3	of	of	ADP
cana-671	6	4	physics	physics	NOUN
cana-671	6	5	and	and	CCONJ
cana-671	6	6	mechanics	mechanic	NOUN
cana-671	6	7	utilize	utilize	VERB
cana-671	6	8	this	this	DET
cana-671	6	9	relation	relation	NOUN
cana-671	6	10	to	to	PART
cana-671	6	11	mathematically	mathematically	ADV
cana-671	6	12	represent	represent	VERB
cana-671	6	13	and	and	CCONJ
cana-671	6	14	address	address	VERB
cana-671	6	15	geometry	geometry	NOUN
cana-671	6	16	-	-	PUNCT
cana-671	6	17	rooted	root	VERB
cana-671	6	18	problems	problem	NOUN
cana-671	6	19	.	.	PUNCT
cana-671	7	1	in	in	ADP
cana-671	7	2	this	this	DET
cana-671	7	3	discussion	discussion	NOUN
cana-671	7	4	,	,	PUNCT
cana-671	7	5	we	we	PRON
cana-671	7	6	will	will	AUX
cana-671	7	7	explore	explore	VERB
cana-671	7	8	various	various	ADJ
cana-671	7	9	categories	category	NOUN
cana-671	7	10	of	of	ADP
cana-671	7	11	geometric	geometric	ADJ
cana-671	7	12	algebra	algebra	NOUN
cana-671	7	13	,	,	PUNCT
cana-671	7	14	delve	delve	VERB
cana-671	7	15	into	into	ADP
cana-671	7	16	the	the	DET
cana-671	7	17	principles	principle	NOUN
cana-671	7	18	of	of	ADP
cana-671	7	19	analysis	analysis	NOUN
cana-671	7	20	,	,	PUNCT
cana-671	7	21	and	and	CCONJ
cana-671	7	22	examine	examine	VERB
cana-671	7	23	the	the	DET
cana-671	7	24	fundamental	fundamental	ADJ
cana-671	7	25	concepts	concept	NOUN
cana-671	7	26	surrounding	surround	VERB
cana-671	7	27	bivectors	bivector	NOUN
cana-671	7	28	,	,	PUNCT
cana-671	7	29	trivectors	trivector	NOUN
cana-671	7	30	,	,	PUNCT
cana-671	7	31	and	and	CCONJ
cana-671	7	32	multivectors	multivector	NOUN
cana-671	7	33	.	.	PUNCT
cana-671	8	1	in	in	ADP
cana-671	8	2	this	this	DET
cana-671	8	3	section	section	NOUN
cana-671	8	4	,	,	PUNCT
cana-671	8	5	we	we	PRON
cana-671	8	6	will	will	AUX
cana-671	8	7	discuss	discuss	VERB
cana-671	8	8	the	the	DET
cana-671	8	9	application	application	NOUN
cana-671	8	10	of	of	ADP
cana-671	8	11	geometric	geometric	ADJ
cana-671	8	12	algebra	algebra	NOUN
cana-671	8	13	in	in	ADP
cana-671	8	14	physics	physics	NOUN
cana-671	8	15	and	and	CCONJ
cana-671	8	16	mechanics	mechanic	NOUN
cana-671	8	17	,	,	PUNCT
cana-671	8	18	specifically	specifically	ADV
cana-671	8	19	focusing	focus	VERB
cana-671	8	20	on	on	ADP
cana-671	8	21	the	the	DET
cana-671	8	22	geometric	geometric	ADJ
cana-671	8	23	algebra	algebra	NOUN
cana-671	8	24	of	of	ADP
cana-671	8	25	3	3	NUM
cana-671	8	26	-	-	PUNCT
cana-671	8	27	d	d	NOUN
cana-671	8	28	space	space	NOUN
cana-671	8	29	,	,	PUNCT
cana-671	8	30	including	include	VERB
cana-671	8	31	rotations	rotation	NOUN
cana-671	8	32	,	,	PUNCT
cana-671	8	33	reflections	reflection	NOUN
cana-671	8	34	,	,	PUNCT
cana-671	8	35	and	and	CCONJ
cana-671	8	36	other	other	ADJ
cana-671	8	37	related	related	ADJ
cana-671	8	38	concepts	concept	NOUN
cana-671	8	39	.	.	PUNCT
cana-671	9	1	by	by	ADP
cana-671	9	2	doing	do	VERB
cana-671	9	3	this	this	PRON
cana-671	9	4	,	,	PUNCT
cana-671	9	5	we	we	PRON
cana-671	9	6	will	will	AUX
cana-671	9	7	develop	develop	VERB
cana-671	9	8	a	a	DET
cana-671	9	9	thorough	thorough	ADJ
cana-671	9	10	understanding	understanding	NOUN
cana-671	9	11	of	of	ADP
cana-671	9	12	how	how	SCONJ
cana-671	9	13	we	we	PRON
cana-671	9	14	can	can	AUX
cana-671	9	15	apply	apply	VERB
cana-671	9	16	geometric	geometric	ADJ
cana-671	9	17	algebra	algebra	NOUN
cana-671	9	18	to	to	PART
cana-671	9	19	resolve	resolve	VERB
cana-671	9	20	fascinating	fascinating	ADJ
cana-671	9	21	physical	physical	ADJ
cana-671	9	22	problems	problem	NOUN
cana-671	9	23	in	in	ADP
cana-671	9	24	the	the	DET
cana-671	9	25	fields	field	NOUN
cana-671	9	26	of	of	ADP
cana-671	9	27	geometry	geometry	NOUN
cana-671	9	28	and	and	CCONJ
cana-671	9	29	mechanics	mechanic	NOUN
cana-671	9	30	.	.	PUNCT
cana-671	10	1	keywords	keyword	NOUN
cana-671	10	2	:	:	PUNCT
cana-671	10	3	algebra	algebra	NOUN
cana-671	10	4	,	,	PUNCT
cana-671	10	5	vector	vector	NOUN
cana-671	10	6	space	space	NOUN
cana-671	10	7	,	,	PUNCT
cana-671	10	8	vector	vector	NOUN
cana-671	10	9	subspace	subspace	NOUN
cana-671	10	10	,	,	PUNCT
cana-671	10	11	vector	vector	NOUN
cana-671	10	12	,	,	PUNCT
cana-671	10	13	bivector	bivector	NOUN
cana-671	10	14	,	,	PUNCT
cana-671	10	15	multivector	multivector	NOUN
cana-671	10	16	.	.	PUNCT
cana-671	11	1	1	1	X
cana-671	11	2	.	.	X
cana-671	11	3	introduction	introduction	NOUN
cana-671	11	4	geometric	geometric	ADJ
cana-671	11	5	algebra	algebra	NOUN
cana-671	11	6	plays	play	VERB
cana-671	11	7	a	a	DET
cana-671	11	8	crucial	crucial	ADJ
cana-671	11	9	role	role	NOUN
cana-671	11	10	in	in	ADP
cana-671	11	11	facilitating	facilitate	VERB
cana-671	11	12	the	the	DET
cana-671	11	13	division	division	NOUN
cana-671	11	14	and	and	CCONJ
cana-671	11	15	addition	addition	NOUN
cana-671	11	16	of	of	ADP
cana-671	11	17	vectors	vector	NOUN
cana-671	11	18	belonging	belong	VERB
cana-671	11	19	to	to	ADP
cana-671	11	20	objects	object	NOUN
cana-671	11	21	with	with	ADP
cana-671	11	22	varying	vary	VERB
cana-671	11	23	dimensions	dimension	NOUN
cana-671	11	24	.	.	PUNCT
cana-671	12	1	clifford	clifford	PROPN
cana-671	12	2	algebra	algebra	PROPN
cana-671	12	3	provides	provide	VERB
cana-671	12	4	a	a	DET
cana-671	12	5	comprehensive	comprehensive	ADJ
cana-671	12	6	framework	framework	NOUN
cana-671	12	7	that	that	PRON
cana-671	12	8	encompasses	encompass	VERB
cana-671	12	9	a	a	DET
cana-671	12	10	significant	significant	ADJ
cana-671	12	11	portion	portion	NOUN
cana-671	12	12	of	of	ADP
cana-671	12	13	mathematics	mathematic	NOUN
cana-671	12	14	,	,	PUNCT
cana-671	12	15	physics	physics	NOUN
cana-671	12	16	,	,	PUNCT
cana-671	12	17	and	and	CCONJ
cana-671	12	18	mechanics	mechanic	NOUN
cana-671	12	19	.	.	PUNCT
cana-671	13	1	it	it	PRON
cana-671	13	2	is	be	AUX
cana-671	13	3	receiving	receive	VERB
cana-671	13	4	growing	grow	VERB
cana-671	13	5	attention	attention	NOUN
cana-671	13	6	in	in	ADP
cana-671	13	7	various	various	ADJ
cana-671	13	8	study	study	NOUN
cana-671	13	9	domains	domain	NOUN
cana-671	13	10	,	,	PUNCT
cana-671	13	11	such	such	ADJ
cana-671	13	12	as	as	ADP
cana-671	13	13	physics	physics	NOUN
cana-671	13	14	,	,	PUNCT
cana-671	13	15	robotics	robotic	NOUN
cana-671	13	16	,	,	PUNCT
cana-671	13	17	and	and	CCONJ
cana-671	13	18	computer	computer	NOUN
cana-671	13	19	graphics	graphic	NOUN
cana-671	13	20	.	.	PUNCT
cana-671	14	1	there	there	PRON
cana-671	14	2	are	be	VERB
cana-671	14	3	an	an	DET
cana-671	14	4	ample	ample	ADJ
cana-671	14	5	number	number	NOUN
cana-671	14	6	of	of	ADP
cana-671	14	7	hypotheses	hypothesis	NOUN
cana-671	14	8	for	for	ADP
cana-671	14	9	this	this	DET
cana-671	14	10	work	work	NOUN
cana-671	14	11	.	.	PUNCT
cana-671	15	1	the	the	DET
cana-671	15	2	interpretation	interpretation	NOUN
cana-671	15	3	of	of	ADP
cana-671	15	4	these	these	DET
cana-671	15	5	theories	theory	NOUN
cana-671	15	6	necessitates	necessitate	VERB
cana-671	15	7	the	the	DET
cana-671	15	8	use	use	NOUN
cana-671	15	9	of	of	ADP
cana-671	15	10	technology	technology	NOUN
cana-671	15	11	,	,	PUNCT
cana-671	15	12	particularly	particularly	ADV
cana-671	15	13	in	in	ADP
cana-671	15	14	the	the	DET
cana-671	15	15	analysis	analysis	NOUN
cana-671	15	16	of	of	ADP
cana-671	15	17	figures	figure	NOUN
cana-671	15	18	and	and	CCONJ
cana-671	15	19	graphs	graph	NOUN
cana-671	15	20	.	.	PUNCT
cana-671	16	1	clifford	clifford	PROPN
cana-671	16	2	algebra	algebra	PROPN
cana-671	16	3	is	be	AUX
cana-671	16	4	a	a	DET
cana-671	16	5	significant	significant	ADJ
cana-671	16	6	theory	theory	NOUN
cana-671	16	7	that	that	PRON
cana-671	16	8	serves	serve	VERB
cana-671	16	9	as	as	ADP
cana-671	16	10	a	a	DET
cana-671	16	11	potent	potent	ADJ
cana-671	16	12	mathematical	mathematical	ADJ
cana-671	16	13	instrument	instrument	NOUN
cana-671	16	14	for	for	ADP
cana-671	16	15	representing	represent	VERB
cana-671	16	16	geometric	geometric	ADJ
cana-671	16	17	objects	object	NOUN
cana-671	16	18	and	and	CCONJ
cana-671	16	19	their	their	PRON
cana-671	16	20	transformations	transformation	NOUN
cana-671	16	21	in	in	ADP
cana-671	16	22	a	a	DET
cana-671	16	23	straightforward	straightforward	ADJ
cana-671	16	24	and	and	CCONJ
cana-671	16	25	intuitive	intuitive	ADJ
cana-671	16	26	manner	manner	NOUN
cana-671	16	27	.	.	PUNCT
cana-671	17	1	clifford	clifford	PROPN
cana-671	17	2	's	's	PART
cana-671	17	3	geometric	geometric	ADJ
cana-671	17	4	algebra	algebra	NOUN
cana-671	17	5	is	be	AUX
cana-671	17	6	an	an	DET
cana-671	17	7	optimal	optimal	ADJ
cana-671	17	8	linguistic	linguistic	ADJ
cana-671	17	9	tool	tool	NOUN
cana-671	17	10	for	for	ADP
cana-671	17	11	physics	physics	NOUN
cana-671	17	12	due	due	ADP
cana-671	17	13	to	to	ADP
cana-671	17	14	its	its	PRON
cana-671	17	15	utilization	utilization	NOUN
cana-671	17	16	of	of	ADP
cana-671	17	17	geometric	geometric	ADJ
cana-671	17	18	properties	property	NOUN
cana-671	17	19	and	and	CCONJ
cana-671	17	20	symmetries	symmetry	NOUN
cana-671	17	21	.	.	PUNCT
cana-671	18	1	2	2	X
cana-671	18	2	.	.	X
cana-671	18	3	geeometr	geeometr	PROPN
cana-671	18	4	’s	’s	PART
cana-671	18	5	role	role	NOUN
cana-671	18	6	in	in	ADP
cana-671	18	7	elementary	elementary	ADJ
cana-671	18	8	algebra	algebra	NOUN
cana-671	18	9	and	and	CCONJ
cana-671	18	10	subspaces	subspace	VERB
cana-671	18	11	geometric	geometric	ADJ
cana-671	18	12	algebra	algebra	NOUN
cana-671	18	13	expands	expand	VERB
cana-671	18	14	upon	upon	SCONJ
cana-671	18	15	the	the	DET
cana-671	18	16	fundamental	fundamental	ADJ
cana-671	18	17	principles	principle	NOUN
cana-671	18	18	of	of	ADP
cana-671	18	19	linear	linear	PROPN
cana-671	18	20	algebra	algebra	NOUN
cana-671	18	21	,	,	PUNCT
cana-671	18	22	including	include	VERB
cana-671	18	23	scalars	scalar	NOUN
cana-671	18	24	and	and	CCONJ
cana-671	18	25	vectors	vector	NOUN
cana-671	18	26	,	,	PUNCT
cana-671	18	27	as	as	ADV
cana-671	18	28	well	well	ADV
cana-671	18	29	as	as	ADP
cana-671	18	30	subspaces	subspace	NOUN
cana-671	18	31	of	of	ADP
cana-671	18	32	two	two	NUM
cana-671	18	33	,	,	PUNCT
cana-671	18	34	three	three	NUM
cana-671	18	35	,	,	PUNCT
cana-671	18	36	or	or	CCONJ
cana-671	18	37	greater	great	ADJ
cana-671	18	38	dimensions	dimension	NOUN
cana-671	18	39	.	.	PUNCT
cana-671	19	1	additionally	additionally	ADV
cana-671	19	2	,	,	PUNCT
cana-671	19	3	it	it	PRON
cana-671	19	4	presents	present	VERB
cana-671	19	5	operators	operator	NOUN
cana-671	19	6	that	that	PRON
cana-671	19	7	facilitate	facilitate	VERB
cana-671	19	8	the	the	DET
cana-671	19	9	execution	execution	NOUN
cana-671	19	10	of	of	ADP
cana-671	19	11	calculations	calculation	NOUN
cana-671	19	12	on	on	ADP
cana-671	19	13	these	these	DET
cana-671	19	14	mathematical	mathematical	ADJ
cana-671	19	15	entities	entity	NOUN
cana-671	19	16	.	.	PUNCT
cana-671	20	1	it	it	PRON
cana-671	20	2	is	be	AUX
cana-671	20	3	indeed	indeed	ADV
cana-671	20	4	possible	possible	ADJ
cana-671	20	5	to	to	PART
cana-671	20	6	combine	combine	VERB
cana-671	20	7	and	and	CCONJ
cana-671	20	8	remove	remove	VERB
cana-671	20	9	subspaces	subspace	NOUN
cana-671	20	10	of	of	ADP
cana-671	20	11	varying	vary	VERB
cana-671	20	12	dimensions	dimension	NOUN
cana-671	20	13	,	,	PUNCT
cana-671	20	14	as	as	ADV
cana-671	20	15	well	well	ADV
cana-671	20	16	as	as	ADP
cana-671	20	17	multiply	multiply	ADP
cana-671	20	18	them	they	PRON
cana-671	20	19	,	,	PUNCT
cana-671	20	20	resulting	result	VERB
cana-671	20	21	in	in	ADP
cana-671	20	22	potent	potent	ADJ
cana-671	20	23	expressions	expression	NOUN
cana-671	20	24	that	that	PRON
cana-671	20	25	can	can	AUX
cana-671	20	26	represent	represent	VERB
cana-671	20	27	any	any	DET
cana-671	20	28	geometric	geometric	ADJ
cana-671	20	29	relationship	relationship	NOUN
cana-671	20	30	or	or	CCONJ
cana-671	20	31	concept	concept	NOUN
cana-671	20	32	.	.	PUNCT
cana-671	21	1	therefore	therefore	ADV
cana-671	21	2	,	,	PUNCT
cana-671	21	3	a	a	DET
cana-671	21	4	significant	significant	ADJ
cana-671	21	5	amount	amount	NOUN
cana-671	21	6	communications	communication	NOUN
cana-671	21	7	on	on	ADP
cana-671	21	8	applied	apply	VERB
cana-671	21	9	nonlinear	nonlinear	ADJ
cana-671	21	10	analysis	analysis	NOUN
cana-671	21	11	issn	issn	NOUN
cana-671	21	12	:	:	PUNCT
cana-671	21	13	1074	1074	NUM
cana-671	21	14	-	-	PUNCT
cana-671	21	15	133x	133x	NUM
cana-671	21	16	vol	vol	NOUN
cana-671	21	17	31	31	NUM
cana-671	21	18	no	no	NOUN
cana-671	21	19	.	.	PUNCT
cana-671	22	1	2s	2s	NUM
cana-671	22	2	(	(	PUNCT
cana-671	22	3	2024	2024	NUM
cana-671	22	4	)	)	PUNCT
cana-671	22	5	573	573	NUM
cana-671	22	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-671	22	7	of	of	ADP
cana-671	22	8	work	work	NOUN
cana-671	22	9	remains	remain	VERB
cana-671	22	10	to	to	PART
cana-671	22	11	discover	discover	VERB
cana-671	22	12	optimal	optimal	ADJ
cana-671	22	13	efficiency	efficiency	NOUN
cana-671	22	14	in	in	ADP
cana-671	22	15	the	the	DET
cana-671	22	16	applications	application	NOUN
cana-671	22	17	of	of	ADP
cana-671	22	18	geometric	geometric	ADJ
cana-671	22	19	algebra	algebra	NOUN
cana-671	22	20	(	(	PUNCT
cana-671	22	21	clifford	clifford	PROPN
cana-671	22	22	)	)	PUNCT
cana-671	22	23	.	.	PUNCT
cana-671	23	1	clifford	clifford	PROPN
cana-671	23	2	algebra	algebra	PROPN
cana-671	23	3	,	,	PUNCT
cana-671	23	4	often	often	ADV
cana-671	23	5	known	know	VERB
cana-671	23	6	as	as	ADP
cana-671	23	7	geometric	geometric	ADJ
cana-671	23	8	algebra	algebra	NOUN
cana-671	23	9	,	,	PUNCT
cana-671	23	10	is	be	AUX
cana-671	23	11	a	a	DET
cana-671	23	12	mathematical	mathematical	ADJ
cana-671	23	13	framework	framework	NOUN
cana-671	23	14	that	that	PRON
cana-671	23	15	expands	expand	VERB
cana-671	23	16	the	the	DET
cana-671	23	17	principles	principle	NOUN
cana-671	23	18	of	of	ADP
cana-671	23	19	vector	vector	NOUN
cana-671	23	20	algebra	algebra	NOUN
cana-671	23	21	to	to	PART
cana-671	23	22	encompass	encompass	VERB
cana-671	23	23	objects	object	NOUN
cana-671	23	24	with	with	ADP
cana-671	23	25	greater	great	ADJ
cana-671	23	26	dimensions	dimension	NOUN
cana-671	23	27	.	.	PUNCT
cana-671	24	1	geometric	geometric	ADJ
cana-671	24	2	algebra	algebra	PROPN
cana-671	24	3	uses	use	VERB
cana-671	24	4	the	the	DET
cana-671	24	5	notion	notion	NOUN
cana-671	24	6	of	of	ADP
cana-671	24	7	multivectors	multivector	NOUN
cana-671	24	8	to	to	PART
cana-671	24	9	depict	depict	VERB
cana-671	24	10	these	these	DET
cana-671	24	11	entities	entity	NOUN
cana-671	24	12	in	in	ADP
cana-671	24	13	dimensions	dimension	NOUN
cana-671	24	14	beyond	beyond	ADP
cana-671	24	15	the	the	DET
cana-671	24	16	usual	usual	ADJ
cana-671	24	17	three	three	NUM
cana-671	24	18	.	.	PUNCT
cana-671	25	1	the	the	DET
cana-671	25	2	zero	zero	NUM
cana-671	25	3	-	-	PUNCT
cana-671	25	4	dimensional	dimensional	ADJ
cana-671	25	5	clifford	clifford	PROPN
cana-671	25	6	algebra	algebra	PROPN
cana-671	25	7	,	,	PUNCT
cana-671	25	8	cl_0	cl_0	PROPN
cana-671	25	9	,	,	PUNCT
cana-671	25	10	is	be	AUX
cana-671	25	11	equivalent	equivalent	ADJ
cana-671	25	12	to	to	ADP
cana-671	25	13	the	the	DET
cana-671	25	14	scalar	scalar	ADJ
cana-671	25	15	algebra	algebra	NOUN
cana-671	25	16	.	.	PUNCT
cana-671	26	1	the	the	DET
cana-671	26	2	clifford	clifford	PROPN
cana-671	26	3	algebra	algebra	PROPN
cana-671	26	4	cl_1	cl_1	ADV
cana-671	26	5	is	be	AUX
cana-671	26	6	a	a	DET
cana-671	26	7	one	one	NUM
cana-671	26	8	-	-	PUNCT
cana-671	26	9	dimensional	dimensional	ADJ
cana-671	26	10	algebraic	algebraic	ADJ
cana-671	26	11	structure	structure	NOUN
cana-671	26	12	consisting	consist	VERB
cana-671	26	13	of	of	ADP
cana-671	26	14	a	a	DET
cana-671	26	15	scalar	scalar	NOUN
cana-671	26	16	and	and	CCONJ
cana-671	26	17	a	a	DET
cana-671	26	18	vector	vector	NOUN
cana-671	26	19	.	.	PUNCT
cana-671	27	1	the	the	DET
cana-671	27	2	two	two	NUM
cana-671	27	3	-	-	PUNCT
cana-671	27	4	dimensional	dimensional	ADJ
cana-671	27	5	clifford	clifford	PROPN
cana-671	27	6	algebra	algebra	PROPN
cana-671	27	7	,	,	PUNCT
cana-671	27	8	cl_2	cl_2	PROPN
cana-671	27	9	,	,	PUNCT
cana-671	27	10	has	have	VERB
cana-671	27	11	a	a	DET
cana-671	27	12	scalar	scalar	ADJ
cana-671	27	13	,	,	PUNCT
cana-671	27	14	two	two	NUM
cana-671	27	15	vectors	vector	NOUN
cana-671	27	16	,	,	PUNCT
cana-671	27	17	and	and	CCONJ
cana-671	27	18	a	a	DET
cana-671	27	19	bivector	bivector	NOUN
cana-671	27	20	.	.	PUNCT
cana-671	28	1	scalar	scalar	ADJ
cana-671	28	2	multiplication	multiplication	NOUN
cana-671	28	3	and	and	CCONJ
cana-671	28	4	vector	vector	NOUN
cana-671	28	5	addition	addition	NOUN
cana-671	28	6	operations	operation	NOUN
cana-671	28	7	can	can	AUX
cana-671	28	8	apply	apply	VERB
cana-671	28	9	to	to	ADP
cana-671	28	10	a	a	DET
cana-671	28	11	vector	vector	NOUN
cana-671	28	12	space	space	NOUN
cana-671	28	13	,	,	PUNCT
cana-671	28	14	a	a	DET
cana-671	28	15	mathematical	mathematical	ADJ
cana-671	28	16	construct	construct	NOUN
cana-671	28	17	that	that	PRON
cana-671	28	18	includes	include	VERB
cana-671	28	19	a	a	DET
cana-671	28	20	collection	collection	NOUN
cana-671	28	21	of	of	ADP
cana-671	28	22	entities	entity	NOUN
cana-671	28	23	known	know	VERB
cana-671	28	24	as	as	ADP
cana-671	28	25	vectors	vector	NOUN
cana-671	28	26	.	.	PUNCT
cana-671	29	1	a	a	DET
cana-671	29	2	subspace	subspace	NOUN
cana-671	29	3	is	be	AUX
cana-671	29	4	a	a	DET
cana-671	29	5	subset	subset	NOUN
cana-671	29	6	of	of	ADP
cana-671	29	7	a	a	DET
cana-671	29	8	vector	vector	NOUN
cana-671	29	9	space	space	NOUN
cana-671	29	10	that	that	PRON
cana-671	29	11	satisfies	satisfy	VERB
cana-671	29	12	closure	closure	NOUN
cana-671	29	13	under	under	ADP
cana-671	29	14	vector	vector	NOUN
cana-671	29	15	addition	addition	NOUN
cana-671	29	16	and	and	CCONJ
cana-671	29	17	scalar	scalar	ADJ
cana-671	29	18	multiplication	multiplication	NOUN
cana-671	29	19	operations	operation	NOUN
cana-671	29	20	.	.	PUNCT
cana-671	30	1	subspaces	subspace	NOUN
cana-671	30	2	are	be	AUX
cana-671	30	3	an	an	DET
cana-671	30	4	important	important	ADJ
cana-671	30	5	concept	concept	NOUN
cana-671	30	6	in	in	ADP
cana-671	30	7	linear	linear	PROPN
cana-671	30	8	algebra	algebra	NOUN
cana-671	30	9	because	because	SCONJ
cana-671	30	10	they	they	PRON
cana-671	30	11	allow	allow	VERB
cana-671	30	12	us	we	PRON
cana-671	30	13	to	to	PART
cana-671	30	14	understand	understand	VERB
cana-671	30	15	the	the	DET
cana-671	30	16	organization	organization	NOUN
cana-671	30	17	of	of	ADP
cana-671	30	18	a	a	DET
cana-671	30	19	vector	vector	NOUN
cana-671	30	20	space	space	NOUN
cana-671	30	21	and	and	CCONJ
cana-671	30	22	tackle	tackle	NOUN
cana-671	30	23	difficulties	difficulty	NOUN
cana-671	30	24	related	relate	VERB
cana-671	30	25	to	to	ADP
cana-671	30	26	linear	linear	ADJ
cana-671	30	27	equations	equation	NOUN
cana-671	30	28	and	and	CCONJ
cana-671	30	29	transformations	transformation	NOUN
cana-671	30	30	.	.	PUNCT
cana-671	31	1	subspaces	subspace	NOUN
cana-671	31	2	are	be	AUX
cana-671	31	3	crucial	crucial	ADJ
cana-671	31	4	in	in	ADP
cana-671	31	5	various	various	ADJ
cana-671	31	6	domains	domain	NOUN
cana-671	31	7	of	of	ADP
cana-671	31	8	mathematics	mathematic	NOUN
cana-671	31	9	and	and	CCONJ
cana-671	31	10	physics	physic	NOUN
cana-671	31	11	,	,	PUNCT
cana-671	31	12	notably	notably	ADV
cana-671	31	13	quantum	quantum	ADJ
cana-671	31	14	mechanics	mechanic	NOUN
cana-671	31	15	.	.	PUNCT
cana-671	32	1	a	a	DET
cana-671	32	2	1	1	NUM
cana-671	32	3	-	-	PUNCT
cana-671	32	4	dimensional	dimensional	ADJ
cana-671	32	5	subspace	subspace	NOUN
cana-671	32	6	can	can	AUX
cana-671	32	7	be	be	AUX
cana-671	32	8	defined	define	VERB
cana-671	32	9	as	as	ADP
cana-671	32	10	a	a	DET
cana-671	32	11	straight	straight	ADJ
cana-671	32	12	line	line	NOUN
cana-671	32	13	that	that	PRON
cana-671	32	14	goes	go	VERB
cana-671	32	15	through	through	ADP
cana-671	32	16	the	the	DET
cana-671	32	17	origin	origin	NOUN
cana-671	32	18	in	in	ADP
cana-671	32	19	a	a	DET
cana-671	32	20	vector	vector	NOUN
cana-671	32	21	space	space	NOUN
cana-671	32	22	.	.	PUNCT
cana-671	33	1	a	a	DET
cana-671	33	2	2	2	NUM
cana-671	33	3	-	-	PUNCT
cana-671	33	4	dimensional	dimensional	ADJ
cana-671	33	5	subspace	subspace	NOUN
cana-671	33	6	refers	refer	VERB
cana-671	33	7	to	to	ADP
cana-671	33	8	a	a	DET
cana-671	33	9	plane	plane	NOUN
cana-671	33	10	that	that	PRON
cana-671	33	11	intersects	intersect	VERB
cana-671	33	12	the	the	DET
cana-671	33	13	origin	origin	NOUN
cana-671	33	14	within	within	ADP
cana-671	33	15	a	a	DET
cana-671	33	16	vector	vector	NOUN
cana-671	33	17	space	space	NOUN
cana-671	33	18	.	.	PUNCT
cana-671	34	1	we	we	PRON
cana-671	34	2	can	can	AUX
cana-671	34	3	express	express	VERB
cana-671	34	4	it	it	PRON
cana-671	34	5	as	as	ADP
cana-671	34	6	a	a	DET
cana-671	34	7	linear	linear	ADJ
cana-671	34	8	combination	combination	NOUN
cana-671	34	9	of	of	ADP
cana-671	34	10	two	two	NUM
cana-671	34	11	vectors	vector	NOUN
cana-671	34	12	that	that	PRON
cana-671	34	13	are	be	AUX
cana-671	34	14	independent	independent	ADJ
cana-671	34	15	of	of	ADP
cana-671	34	16	each	each	DET
cana-671	34	17	other	other	ADJ
cana-671	34	18	.	.	PUNCT
cana-671	35	1	the	the	DET
cana-671	35	2	x	x	NOUN
cana-671	35	3	,	,	PUNCT
cana-671	35	4	y	y	PROPN
cana-671	35	5	plane	plane	NOUN
cana-671	35	6	is	be	AUX
cana-671	35	7	an	an	DET
cana-671	35	8	example	example	NOUN
cana-671	35	9	of	of	ADP
cana-671	35	10	a	a	DET
cana-671	35	11	one	one	NUM
cana-671	35	12	-	-	PUNCT
cana-671	35	13	dimensional	dimensional	ADJ
cana-671	35	14	subspace	subspace	NOUN
cana-671	35	15	in	in	ADP
cana-671	35	16	a	a	DET
cana-671	35	17	three	three	NUM
cana-671	35	18	-	-	PUNCT
cana-671	35	19	dimensional	dimensional	ADJ
cana-671	35	20	space	space	NOUN
cana-671	35	21	.	.	PUNCT
cana-671	36	1	a	a	DET
cana-671	36	2	three	three	NUM
cana-671	36	3	-	-	PUNCT
cana-671	36	4	dimensional	dimensional	ADJ
cana-671	36	5	subspace	subspace	NOUN
cana-671	36	6	encompasses	encompass	VERB
cana-671	36	7	the	the	DET
cana-671	36	8	entirety	entirety	NOUN
cana-671	36	9	of	of	ADP
cana-671	36	10	the	the	DET
cana-671	36	11	vector	vector	NOUN
cana-671	36	12	space	space	NOUN
cana-671	36	13	.	.	PUNCT
cana-671	37	1	for	for	ADP
cana-671	37	2	example	example	NOUN
cana-671	37	3	,	,	PUNCT
cana-671	37	4	a	a	DET
cana-671	37	5	3	3	NUM
cana-671	37	6	-	-	PUNCT
cana-671	37	7	dimensional	dimensional	ADJ
cana-671	37	8	space	space	NOUN
cana-671	37	9	can	can	AUX
cana-671	37	10	be	be	AUX
cana-671	37	11	considered	consider	VERB
cana-671	37	12	a	a	DET
cana-671	37	13	subspace	subspace	NOUN
cana-671	37	14	within	within	ADP
cana-671	37	15	a	a	DET
cana-671	37	16	3	3	NUM
cana-671	37	17	-	-	PUNCT
cana-671	37	18	dimensional	dimensional	ADJ
cana-671	37	19	framework	framework	NOUN
cana-671	37	20	.	.	PUNCT
cana-671	38	1	we	we	PRON
cana-671	38	2	can	can	AUX
cana-671	38	3	view	view	VERB
cana-671	38	4	scalars	scalar	NOUN
cana-671	38	5	as	as	ADP
cana-671	38	6	subspaces	subspace	NOUN
cana-671	38	7	with	with	ADP
cana-671	38	8	zero	zero	NUM
cana-671	38	9	dimensions	dimension	NOUN
cana-671	38	10	.	.	PUNCT
cana-671	39	1	3.algebra	3.algebra	NUM
cana-671	39	2	through	through	ADP
cana-671	39	3	geometry	geometry	NOUN
cana-671	39	4	a	a	DET
cana-671	39	5	geometric	geometric	ADJ
cana-671	39	6	algebra	algebra	NOUN
cana-671	39	7	,	,	PUNCT
cana-671	39	8	also	also	ADV
cana-671	39	9	referred	refer	VERB
cana-671	39	10	to	to	ADP
cana-671	39	11	as	as	ADP
cana-671	39	12	a	a	DET
cana-671	39	13	true	true	ADJ
cana-671	39	14	clifford	clifford	PROPN
cana-671	39	15	algebra	algebra	PROPN
cana-671	39	16	,	,	PUNCT
cana-671	39	17	is	be	AUX
cana-671	39	18	a	a	DET
cana-671	39	19	mathematical	mathematical	ADJ
cana-671	39	20	extension	extension	NOUN
cana-671	39	21	of	of	ADP
cana-671	39	22	elementary	elementary	ADJ
cana-671	39	23	algebra	algebra	PROPN
cana-671	39	24	that	that	PRON
cana-671	39	25	allows	allow	VERB
cana-671	39	26	for	for	ADP
cana-671	39	27	the	the	DET
cana-671	39	28	manipulation	manipulation	NOUN
cana-671	39	29	of	of	ADP
cana-671	39	30	geometric	geometric	ADJ
cana-671	39	31	entities	entity	NOUN
cana-671	39	32	,	,	PUNCT
cana-671	39	33	such	such	ADJ
cana-671	39	34	as	as	ADP
cana-671	39	35	vectors	vector	NOUN
cana-671	39	36	.	.	PUNCT
cana-671	40	1	two	two	NUM
cana-671	40	2	fundamental	fundamental	ADJ
cana-671	40	3	operations	operation	NOUN
cana-671	40	4	,	,	PUNCT
cana-671	40	5	addition	addition	NOUN
cana-671	40	6	and	and	CCONJ
cana-671	40	7	geometric	geometric	ADJ
cana-671	40	8	multiplication	multiplication	NOUN
cana-671	40	9	,	,	PUNCT
cana-671	40	10	construct	construct	VERB
cana-671	40	11	geometric	geometric	ADJ
cana-671	40	12	algebra	algebra	NOUN
cana-671	40	13	.	.	PUNCT
cana-671	41	1	the	the	DET
cana-671	41	2	multiplication	multiplication	NOUN
cana-671	41	3	of	of	ADP
cana-671	41	4	vectors	vector	NOUN
cana-671	41	5	yields	yield	VERB
cana-671	41	6	multivectors	multivector	NOUN
cana-671	41	7	,	,	PUNCT
cana-671	41	8	which	which	PRON
cana-671	41	9	are	be	AUX
cana-671	41	10	objects	object	NOUN
cana-671	41	11	with	with	ADP
cana-671	41	12	higher	high	ADJ
cana-671	41	13	dimensions	dimension	NOUN
cana-671	41	14	.	.	PUNCT
cana-671	42	1	geometric	geometric	ADJ
cana-671	42	2	algebra	algebra	NOUN
cana-671	42	3	is	be	AUX
cana-671	42	4	crucial	crucial	ADJ
cana-671	42	5	in	in	ADP
cana-671	42	6	facilitating	facilitate	VERB
cana-671	42	7	vector	vector	NOUN
cana-671	42	8	division	division	NOUN
cana-671	42	9	and	and	CCONJ
cana-671	42	10	addition	addition	NOUN
cana-671	42	11	of	of	ADP
cana-671	42	12	objects	object	NOUN
cana-671	42	13	with	with	ADP
cana-671	42	14	varying	vary	VERB
cana-671	42	15	dimensions	dimension	NOUN
cana-671	42	16	when	when	SCONJ
cana-671	42	17	compared	compare	VERB
cana-671	42	18	to	to	ADP
cana-671	42	19	other	other	ADJ
cana-671	42	20	methods	method	NOUN
cana-671	42	21	of	of	ADP
cana-671	42	22	handling	handle	VERB
cana-671	42	23	geometric	geometric	ADJ
cana-671	42	24	objects	object	NOUN
cana-671	42	25	.	.	PUNCT
cana-671	43	1	for	for	ADP
cana-671	43	2	many	many	ADJ
cana-671	43	3	years	year	NOUN
cana-671	43	4	,	,	PUNCT
cana-671	43	5	the	the	DET
cana-671	43	6	recently	recently	ADV
cana-671	43	7	created	create	VERB
cana-671	43	8	vector	vector	NOUN
cana-671	43	9	calculus	calculus	NOUN
cana-671	43	10	,	,	PUNCT
cana-671	43	11	which	which	PRON
cana-671	43	12	described	describe	VERB
cana-671	43	13	electromagnetism	electromagnetism	NOUN
cana-671	43	14	,	,	PUNCT
cana-671	43	15	overshadowed	overshadow	VERB
cana-671	43	16	geometric	geometric	ADJ
cana-671	43	17	algebras	algebra	NOUN
cana-671	43	18	.	.	PUNCT
cana-671	44	1	you	you	PRON
cana-671	44	2	can	can	AUX
cana-671	44	3	use	use	VERB
cana-671	44	4	a	a	DET
cana-671	44	5	trivector	trivector	NOUN
cana-671	44	6	to	to	PART
cana-671	44	7	represent	represent	VERB
cana-671	44	8	a	a	DET
cana-671	44	9	volume	volume	NOUN
cana-671	44	10	with	with	ADP
cana-671	44	11	a	a	DET
cana-671	44	12	specific	specific	ADJ
cana-671	44	13	orientation	orientation	NOUN
cana-671	44	14	.	.	PUNCT
cana-671	45	1	elements	element	NOUN
cana-671	45	2	represent	represent	VERB
cana-671	45	3	rotations	rotation	NOUN
cana-671	45	4	and	and	CCONJ
cana-671	45	5	reflections	reflection	NOUN
cana-671	45	6	.	.	PUNCT
cana-671	46	1	in	in	ADP
cana-671	46	2	contrast	contrast	NOUN
cana-671	46	3	to	to	ADP
cana-671	46	4	vector	vector	NOUN
cana-671	46	5	algebra	algebra	NOUN
cana-671	46	6	,	,	PUNCT
cana-671	46	7	geometric	geometric	ADJ
cana-671	46	8	algebra	algebra	NOUN
cana-671	46	9	inherently	inherently	ADV
cana-671	46	10	supports	support	VERB
cana-671	46	11	an	an	DET
cana-671	46	12	unlimited	unlimited	ADJ
cana-671	46	13	number	number	NOUN
cana-671	46	14	of	of	ADP
cana-671	46	15	dimensions	dimension	NOUN
cana-671	46	16	and	and	CCONJ
cana-671	46	17	any	any	DET
cana-671	46	18	quadratic	quadratic	ADJ
cana-671	46	19	form	form	NOUN
cana-671	46	20	,	,	PUNCT
cana-671	46	21	as	as	SCONJ
cana-671	46	22	is	be	AUX
cana-671	46	23	the	the	DET
cana-671	46	24	case	case	NOUN
cana-671	46	25	in	in	ADP
cana-671	46	26	relativity	relativity	NOUN
cana-671	46	27	theory	theory	NOUN
cana-671	46	28	.	.	PUNCT
cana-671	47	1	physics	physics	NOUN
cana-671	47	2	uses	use	VERB
cana-671	47	3	geometric	geometric	ADJ
cana-671	47	4	algebras	algebra	NOUN
cana-671	47	5	like	like	ADP
cana-671	47	6	space	space	NOUN
cana-671	47	7	-	-	PUNCT
cana-671	47	8	time	time	NOUN
cana-671	47	9	algebra	algebra	NOUN
cana-671	47	10	,	,	PUNCT
cana-671	47	11	which	which	PRON
cana-671	47	12	includes	include	VERB
cana-671	47	13	the	the	DET
cana-671	47	14	less	less	ADV
cana-671	47	15	common	common	ADJ
cana-671	47	16	physical	physical	ADJ
cana-671	47	17	space	space	NOUN
cana-671	47	18	algebra	algebra	NOUN
cana-671	47	19	,	,	PUNCT
cana-671	47	20	and	and	CCONJ
cana-671	47	21	conformal	conformal	ADJ
cana-671	47	22	geometric	geometric	ADJ
cana-671	47	23	algebra	algebra	NOUN
cana-671	47	24	.	.	PUNCT
cana-671	48	1	we	we	PRON
cana-671	48	2	can	can	AUX
cana-671	48	3	employ	employ	VERB
cana-671	48	4	geometric	geometric	ADJ
cana-671	48	5	calculus	calculus	NOUN
cana-671	48	6	,	,	PUNCT
cana-671	48	7	an	an	DET
cana-671	48	8	expanded	expand	VERB
cana-671	48	9	version	version	NOUN
cana-671	48	10	of	of	ADP
cana-671	48	11	geometric	geometric	ADJ
cana-671	48	12	algebra	algebra	NOUN
cana-671	48	13	that	that	PRON
cana-671	48	14	encompasses	encompass	VERB
cana-671	48	15	differentiation	differentiation	NOUN
cana-671	48	16	and	and	CCONJ
cana-671	48	17	integration	integration	NOUN
cana-671	48	18	,	,	PUNCT
cana-671	48	19	to	to	PART
cana-671	48	20	develop	develop	VERB
cana-671	48	21	additional	additional	ADJ
cana-671	48	22	theories	theory	NOUN
cana-671	48	23	like	like	ADP
cana-671	48	24	complex	complex	ADJ
cana-671	48	25	analysis	analysis	NOUN
cana-671	48	26	and	and	CCONJ
cana-671	48	27	differential	differential	NOUN
cana-671	48	28	geometry	geometry	NOUN
cana-671	48	29	.	.	PUNCT
cana-671	49	1	we	we	PRON
cana-671	49	2	can	can	AUX
cana-671	49	3	achieve	achieve	VERB
cana-671	49	4	this	this	PRON
cana-671	49	5	by	by	ADP
cana-671	49	6	substituting	substitute	VERB
cana-671	49	7	differential	differential	ADJ
cana-671	49	8	forms	form	NOUN
cana-671	49	9	with	with	ADP
cana-671	49	10	clifford	clifford	PROPN
cana-671	49	11	algebra	algebra	PROPN
cana-671	49	12	.	.	PUNCT
cana-671	50	1	david	david	PROPN
cana-671	50	2	hestenes	hestenes	PROPN
cana-671	50	3	and	and	CCONJ
cana-671	50	4	chris	chris	PROPN
cana-671	50	5	doran	doran	PROPN
cana-671	50	6	,	,	PUNCT
cana-671	50	7	in	in	ADP
cana-671	50	8	particular	particular	ADJ
cana-671	50	9	,	,	PUNCT
cana-671	50	10	have	have	AUX
cana-671	50	11	promoted	promote	VERB
cana-671	50	12	geometric	geometric	ADJ
cana-671	50	13	algebra	algebra	NOUN
cana-671	50	14	as	as	ADP
cana-671	50	15	the	the	DET
cana-671	50	16	preferred	preferred	ADJ
cana-671	50	17	mathematical	mathematical	ADJ
cana-671	50	18	framework	framework	NOUN
cana-671	50	19	for	for	ADP
cana-671	50	20	physics	physic	NOUN
cana-671	50	21	.	.	PUNCT
cana-671	51	1	advocates	advocate	NOUN
cana-671	51	2	argue	argue	VERB
cana-671	51	3	that	that	SCONJ
cana-671	51	4	it	it	PRON
cana-671	51	5	offers	offer	VERB
cana-671	51	6	concise	concise	ADJ
cana-671	51	7	and	and	CCONJ
cana-671	51	8	easy	easy	ADJ
cana-671	51	9	-	-	PUNCT
cana-671	51	10	to	to	PART
cana-671	51	11	-	-	PUNCT
cana-671	51	12	understand	understand	VERB
cana-671	51	13	explanations	explanation	NOUN
cana-671	51	14	of	of	ADP
cana-671	51	15	several	several	ADJ
cana-671	51	16	disciplines	discipline	NOUN
cana-671	51	17	,	,	PUNCT
cana-671	51	18	like	like	ADP
cana-671	51	19	classical	classical	ADJ
cana-671	51	20	and	and	CCONJ
cana-671	51	21	quantum	quantum	NOUN
cana-671	51	22	physics	physics	NOUN
cana-671	51	23	,	,	PUNCT
cana-671	51	24	electromagnetic	electromagnetic	ADJ
cana-671	51	25	theory	theory	NOUN
cana-671	51	26	,	,	PUNCT
cana-671	51	27	and	and	CCONJ
cana-671	51	28	relativity	relativity	NOUN
cana-671	51	29	.	.	PUNCT
cana-671	52	1	the	the	DET
cana-671	52	2	fields	field	NOUN
cana-671	52	3	of	of	ADP
cana-671	52	4	computer	computer	NOUN
cana-671	52	5	graphics	graphic	NOUN
cana-671	52	6	and	and	CCONJ
cana-671	52	7	robotics	robotic	NOUN
cana-671	52	8	employ	employ	VERB
cana-671	52	9	geometric	geometric	ADJ
cana-671	52	10	algebra	algebra	NOUN
cana-671	52	11	as	as	ADP
cana-671	52	12	a	a	DET
cana-671	52	13	computational	computational	ADJ
cana-671	52	14	tool	tool	NOUN
cana-671	52	15	.	.	PUNCT
cana-671	53	1	below	below	ADV
cana-671	53	2	,	,	PUNCT
cana-671	53	3	we	we	PRON
cana-671	53	4	shall	shall	AUX
cana-671	53	5	outline	outline	VERB
cana-671	53	6	several	several	ADJ
cana-671	53	7	key	key	ADJ
cana-671	53	8	aspects	aspect	NOUN
cana-671	53	9	of	of	ADP
cana-671	53	10	subspaces	subspace	NOUN
cana-671	53	11	.	.	PUNCT
cana-671	54	1	communications	communication	NOUN
cana-671	54	2	on	on	ADP
cana-671	54	3	applied	apply	VERB
cana-671	54	4	nonlinear	nonlinear	ADJ
cana-671	54	5	analysis	analysis	NOUN
cana-671	54	6	issn	issn	NOUN
cana-671	54	7	:	:	PUNCT
cana-671	54	8	1074	1074	NUM
cana-671	54	9	-	-	PUNCT
cana-671	54	10	133x	133x	NUM
cana-671	54	11	vol	vol	NOUN
cana-671	54	12	31	31	NUM
cana-671	54	13	no	no	NOUN
cana-671	54	14	.	.	PUNCT
cana-671	55	1	2s	2s	NUM
cana-671	55	2	(	(	PUNCT
cana-671	55	3	2024	2024	NUM
cana-671	55	4	)	)	PUNCT
cana-671	55	5	574	574	NUM
cana-671	55	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-671	55	7	many	many	ADJ
cana-671	55	8	people	people	NOUN
cana-671	55	9	fail	fail	VERB
cana-671	55	10	to	to	PART
cana-671	55	11	recognize	recognize	VERB
cana-671	55	12	that	that	SCONJ
cana-671	55	13	vectors	vector	NOUN
cana-671	55	14	actually	actually	ADV
cana-671	55	15	represent	represent	VERB
cana-671	55	16	subspaces	subspace	NOUN
cana-671	55	17	that	that	PRON
cana-671	55	18	exist	exist	VERB
cana-671	55	19	in	in	ADP
cana-671	55	20	one	one	NUM
cana-671	55	21	dimension	dimension	NOUN
cana-671	55	22	.	.	PUNCT
cana-671	56	1	thus	thus	ADV
cana-671	56	2	,	,	PUNCT
cana-671	56	3	we	we	PRON
cana-671	56	4	employ	employ	VERB
cana-671	56	5	vectors	vector	NOUN
cana-671	56	6	to	to	PART
cana-671	56	7	create	create	VERB
cana-671	56	8	subspaces	subspace	NOUN
cana-671	56	9	of	of	ADP
cana-671	56	10	higher	high	ADJ
cana-671	56	11	dimensions	dimension	NOUN
cana-671	56	12	.	.	PUNCT
cana-671	57	1	planes	plane	NOUN
cana-671	57	2	are	be	AUX
cana-671	57	3	represented	represent	VERB
cana-671	57	4	by	by	ADP
cana-671	57	5	specifying	specify	VERB
cana-671	57	6	normals	normal	NOUN
cana-671	57	7	.	.	PUNCT
cana-671	58	1	we	we	PRON
cana-671	58	2	reject	reject	VERB
cana-671	58	3	the	the	DET
cana-671	58	4	notion	notion	NOUN
cana-671	58	5	that	that	SCONJ
cana-671	58	6	2	2	NUM
cana-671	58	7	,	,	PUNCT
cana-671	58	8	3	3	NUM
cana-671	58	9	,	,	PUNCT
cana-671	58	10	and	and	CCONJ
cana-671	58	11	higher	higher	ADV
cana-671	58	12	-	-	PUNCT
cana-671	58	13	dimensional	dimensional	ADJ
cana-671	58	14	subspaces	subspace	NOUN
cana-671	58	15	are	be	AUX
cana-671	58	16	straightforward	straightforward	ADJ
cana-671	58	17	and	and	CCONJ
cana-671	58	18	analogous	analogous	ADJ
cana-671	58	19	to	to	ADP
cana-671	58	20	vectors	vector	NOUN
cana-671	58	21	.	.	PUNCT
cana-671	59	1	algebraic	algebraic	ADJ
cana-671	59	2	geometry	geometry	NOUN
cana-671	59	3	introduces	introduce	NOUN
cana-671	59	4	and	and	CCONJ
cana-671	59	5	provides	provide	VERB
cana-671	59	6	operators	operator	NOUN
cana-671	59	7	for	for	ADP
cana-671	59	8	doing	do	VERB
cana-671	59	9	arithmetic	arithmetic	NOUN
cana-671	59	10	with	with	ADP
cana-671	59	11	mathematical	mathematical	ADJ
cana-671	59	12	objects	object	NOUN
cana-671	59	13	.	.	PUNCT
cana-671	60	1	geometric	geometric	ADJ
cana-671	60	2	algebra	algebra	NOUN
cana-671	60	3	enables	enable	VERB
cana-671	60	4	us	we	PRON
cana-671	60	5	to	to	PART
cana-671	60	6	accurately	accurately	ADV
cana-671	60	7	depict	depict	VERB
cana-671	60	8	planes	plane	NOUN
cana-671	60	9	as	as	ADV
cana-671	60	10	genuine	genuine	ADJ
cana-671	60	11	2	2	NUM
cana-671	60	12	-	-	PUNCT
cana-671	60	13	dimensional	dimensional	ADJ
cana-671	60	14	subspaces	subspace	NOUN
cana-671	60	15	,	,	PUNCT
cana-671	60	16	establish	establish	VERB
cana-671	60	17	subspace	subspace	NOUN
cana-671	60	18	orientation	orientation	NOUN
cana-671	60	19	,	,	PUNCT
cana-671	60	20	and	and	CCONJ
cana-671	60	21	uncover	uncover	VERB
cana-671	60	22	the	the	DET
cana-671	60	23	authentic	authentic	ADJ
cana-671	60	24	nature	nature	NOUN
cana-671	60	25	of	of	ADP
cana-671	60	26	quaternions	quaternion	NOUN
cana-671	60	27	.	.	PUNCT
cana-671	61	1	we	we	PRON
cana-671	61	2	can	can	AUX
cana-671	61	3	add	add	VERB
cana-671	61	4	,	,	PUNCT
cana-671	61	5	subtract	subtract	ADJ
cana-671	61	6	,	,	PUNCT
cana-671	61	7	multiply	multiply	NOUN
cana-671	61	8	,	,	PUNCT
cana-671	61	9	and	and	CCONJ
cana-671	61	10	divide	divide	VERB
cana-671	61	11	subspaces	subspace	NOUN
cana-671	61	12	of	of	ADP
cana-671	61	13	different	different	ADJ
cana-671	61	14	dimensions	dimension	NOUN
cana-671	61	15	to	to	PART
cana-671	61	16	create	create	VERB
cana-671	61	17	powerful	powerful	ADJ
cana-671	61	18	expressions	expression	NOUN
cana-671	61	19	that	that	PRON
cana-671	61	20	can	can	AUX
cana-671	61	21	represent	represent	VERB
cana-671	61	22	any	any	DET
cana-671	61	23	geometric	geometric	ADJ
cana-671	61	24	connection	connection	NOUN
cana-671	61	25	or	or	CCONJ
cana-671	61	26	concept	concept	NOUN
cana-671	61	27	.	.	PUNCT
cana-671	62	1	this	this	DET
cana-671	62	2	work	work	NOUN
cana-671	62	3	aims	aim	VERB
cana-671	62	4	to	to	PART
cana-671	62	5	illustrate	illustrate	VERB
cana-671	62	6	the	the	DET
cana-671	62	7	representation	representation	NOUN
cana-671	62	8	of	of	ADP
cana-671	62	9	1	1	NUM
cana-671	62	10	-	-	PUNCT
cana-671	62	11	dimensional	dimensional	ADJ
cana-671	62	12	subspaces	subspace	NOUN
cana-671	62	13	using	use	VERB
cana-671	62	14	vectors	vector	NOUN
cana-671	62	15	and	and	CCONJ
cana-671	62	16	utilize	utilize	VERB
cana-671	62	17	this	this	DET
cana-671	62	18	understanding	understanding	NOUN
cana-671	62	19	to	to	PART
cana-671	62	20	describe	describe	VERB
cana-671	62	21	subspaces	subspace	NOUN
cana-671	62	22	of	of	ADP
cana-671	62	23	arbitrary	arbitrary	ADJ
cana-671	62	24	dimensions	dimension	NOUN
cana-671	62	25	.	.	PUNCT
cana-671	63	1	before	before	ADP
cana-671	63	2	proceeding	proceeding	NOUN
cana-671	63	3	,	,	PUNCT
cana-671	63	4	let	let	VERB
cana-671	63	5	us	we	PRON
cana-671	63	6	first	first	ADV
cana-671	63	7	revisit	revisit	VERB
cana-671	63	8	the	the	DET
cana-671	63	9	fundamental	fundamental	ADJ
cana-671	63	10	concepts	concept	NOUN
cana-671	63	11	using	use	VERB
cana-671	63	12	a	a	DET
cana-671	63	13	well	well	ADV
cana-671	63	14	-	-	PUNCT
cana-671	63	15	known	know	VERB
cana-671	63	16	illustration	illustration	NOUN
cana-671	63	17	.	.	PUNCT
cana-671	64	1	figure	figure	NOUN
cana-671	64	2	1	1	NUM
cana-671	64	3	:	:	PUNCT
cana-671	64	4	scalar	scalar	ADJ
cana-671	64	5	product	product	NOUN
cana-671	64	6	what	what	PRON
cana-671	64	7	happens	happen	VERB
cana-671	64	8	when	when	SCONJ
cana-671	64	9	we	we	PRON
cana-671	64	10	project	project	VERB
cana-671	64	11	a	a	DET
cana-671	64	12	one	one	NUM
cana-671	64	13	-	-	PUNCT
cana-671	64	14	dimensional	dimensional	ADJ
cana-671	64	15	subspace	subspace	NOUN
cana-671	64	16	onto	onto	ADP
cana-671	64	17	another	another	PRON
cana-671	64	18	?	?	PUNCT
cana-671	65	1	the	the	DET
cana-671	65	2	solution	solution	NOUN
cana-671	65	3	is	be	AUX
cana-671	65	4	widely	widely	ADV
cana-671	65	5	recognized	recognize	VERB
cana-671	65	6	:	:	PUNCT
cana-671	65	7	when	when	SCONJ
cana-671	65	8	considering	consider	VERB
cana-671	65	9	vectors	vector	NOUN
cana-671	65	10	a	a	PRON
cana-671	65	11	and	and	CCONJ
cana-671	65	12	b	b	NOUN
cana-671	65	13	,	,	PUNCT
cana-671	65	14	the	the	DET
cana-671	65	15	scalar	scalar	ADJ
cana-671	65	16	product	product	NOUN
cana-671	65	17	a⋅b	a⋅b	NOUN
cana-671	65	18	represents	represent	VERB
cana-671	65	19	the	the	DET
cana-671	65	20	projection	projection	NOUN
cana-671	65	21	of	of	ADP
cana-671	65	22	vector	vector	NOUN
cana-671	65	23	an	an	PRON
cana-671	65	24	onto	onto	ADP
cana-671	65	25	vector	vector	NOUN
cana-671	65	26	b	b	NOUN
cana-671	65	27	,	,	PUNCT
cana-671	65	28	yielding	yield	VERB
cana-671	65	29	the	the	DET
cana-671	65	30	scalar	scalar	ADJ
cana-671	65	31	magnitude	magnitude	NOUN
cana-671	65	32	of	of	ADP
cana-671	65	33	the	the	DET
cana-671	65	34	projection	projection	NOUN
cana-671	65	35	in	in	ADP
cana-671	65	36	relation	relation	NOUN
cana-671	65	37	to	to	ADP
cana-671	65	38	the	the	DET
cana-671	65	39	magnitude	magnitude	NOUN
cana-671	65	40	of	of	ADP
cana-671	65	41	b.	b.	PROPN
cana-671	65	42	figure	figure	NOUN
cana-671	65	43	1	1	NUM
cana-671	65	44	illustrates	illustrate	VERB
cana-671	65	45	the	the	DET
cana-671	65	46	scenario	scenario	NOUN
cana-671	65	47	where	where	SCONJ
cana-671	65	48	b	b	NOUN
cana-671	65	49	is	be	AUX
cana-671	65	50	a	a	DET
cana-671	65	51	unit	unit	NOUN
cana-671	65	52	vector	vector	NOUN
cana-671	65	53	.	.	PUNCT
cana-671	66	1	we	we	PRON
cana-671	66	2	can	can	AUX
cana-671	66	3	view	view	VERB
cana-671	66	4	scalars	scalar	NOUN
cana-671	66	5	as	as	ADP
cana-671	66	6	subspaces	subspace	NOUN
cana-671	66	7	with	with	ADP
cana-671	66	8	zero	zero	NUM
cana-671	66	9	dimensions	dimension	NOUN
cana-671	66	10	.	.	PUNCT
cana-671	67	1	therefore	therefore	ADV
cana-671	67	2	,	,	PUNCT
cana-671	67	3	projecting	project	VERB
cana-671	67	4	one	one	NUM
cana-671	67	5	1	1	NUM
cana-671	67	6	-	-	PUNCT
cana-671	67	7	dimensional	dimensional	ADJ
cana-671	67	8	subspace	subspace	NOUN
cana-671	67	9	onto	onto	ADP
cana-671	67	10	another	another	DET
cana-671	67	11	results	result	NOUN
cana-671	67	12	in	in	ADP
cana-671	67	13	a	a	DET
cana-671	67	14	0	0	NUM
cana-671	67	15	-	-	PUNCT
cana-671	67	16	dimensional	dimensional	ADJ
cana-671	67	17	subspace	subspace	NOUN
cana-671	67	18	.	.	PUNCT
cana-671	68	1	4	4	X
cana-671	68	2	.	.	NOUN
cana-671	68	3	three	three	NUM
cana-671	68	4	types	type	NOUN
cana-671	68	5	of	of	ADP
cana-671	68	6	algebra	algebra	NOUN
cana-671	68	7	in	in	ADP
cana-671	68	8	order	order	NOUN
cana-671	68	9	to	to	PART
cana-671	68	10	solve	solve	VERB
cana-671	68	11	the	the	DET
cana-671	68	12	equations	equation	NOUN
cana-671	68	13	,	,	PUNCT
cana-671	68	14	it	it	PRON
cana-671	68	15	is	be	AUX
cana-671	68	16	necessary	necessary	ADJ
cana-671	68	17	to	to	PART
cana-671	68	18	utilize	utilize	VERB
cana-671	68	19	algebraic	algebraic	ADJ
cana-671	68	20	identities	identity	NOUN
cana-671	68	21	,	,	PUNCT
cana-671	68	22	such	such	ADJ
cana-671	68	23	as	as	ADP
cana-671	68	24	(	(	PUNCT
cana-671	68	25	𝑎	𝑎	X
cana-671	68	26	+	+	X
cana-671	68	27	𝑏)2=𝑎2	𝑏)2=𝑎2	PROPN
cana-671	68	28	+	+	CCONJ
cana-671	68	29	𝑏2	𝑏2	PROPN
cana-671	68	30	+	+	CCONJ
cana-671	68	31	2𝑎𝑏	2𝑎𝑏	ADJ
cana-671	68	32	(	(	PUNCT
cana-671	68	33	1	1	NUM
cana-671	68	34	)	)	PUNCT
cana-671	68	35	euclid	euclid	PROPN
cana-671	68	36	's	's	PART
cana-671	68	37	elements	element	NOUN
cana-671	68	38	provides	provide	VERB
cana-671	68	39	a	a	DET
cana-671	68	40	geometric	geometric	ADJ
cana-671	68	41	illustration	illustration	NOUN
cana-671	68	42	of	of	ADP
cana-671	68	43	this	this	DET
cana-671	68	44	identity	identity	NOUN
cana-671	68	45	.	.	PUNCT
cana-671	69	1	divide	divide	VERB
cana-671	69	2	a	a	DET
cana-671	69	3	given	give	VERB
cana-671	69	4	length	length	NOUN
cana-671	69	5	arbitrarily	arbitrarily	ADV
cana-671	69	6	,	,	PUNCT
cana-671	69	7	and	and	CCONJ
cana-671	69	8	the	the	DET
cana-671	69	9	square	square	NOUN
cana-671	69	10	of	of	ADP
cana-671	69	11	the	the	DET
cana-671	69	12	entire	entire	ADJ
cana-671	69	13	length	length	NOUN
cana-671	69	14	equals	equal	VERB
cana-671	69	15	the	the	DET
cana-671	69	16	sum	sum	NOUN
cana-671	69	17	of	of	ADP
cana-671	69	18	the	the	DET
cana-671	69	19	squares	square	NOUN
cana-671	69	20	of	of	ADP
cana-671	69	21	the	the	DET
cana-671	69	22	segments	segment	NOUN
cana-671	69	23	and	and	CCONJ
cana-671	69	24	the	the	DET
cana-671	69	25	double	double	ADJ
cana-671	69	26	rectangle	rectangle	NOUN
cana-671	69	27	the	the	DET
cana-671	69	28	segments	segment	NOUN
cana-671	69	29	form	form	VERB
cana-671	69	30	.	.	PUNCT
cana-671	70	1	al	al	PROPN
cana-671	70	2	-	-	PUNCT
cana-671	70	3	khwarizmi	khwarizmi	PROPN
cana-671	70	4	employs	employ	VERB
cana-671	70	5	the	the	DET
cana-671	70	6	mathematical	mathematical	ADJ
cana-671	70	7	concept	concept	NOUN
cana-671	70	8	of	of	ADP
cana-671	70	9	identity	identity	NOUN
cana-671	70	10	(	(	PUNCT
cana-671	70	11	1	1	NUM
cana-671	70	12	)	)	PUNCT
cana-671	70	13	to	to	PART
cana-671	70	14	solve	solve	VERB
cana-671	70	15	equations	equation	NOUN
cana-671	70	16	with	with	ADP
cana-671	70	17	a	a	DET
cana-671	70	18	specific	specific	ADJ
cana-671	70	19	structure	structure	NOUN
cana-671	70	20	.	.	PUNCT
cana-671	71	1	𝑥2	𝑥2	NOUN
cana-671	71	2	+	+	CCONJ
cana-671	71	3	𝑎𝑥	𝑎𝑥	NOUN
cana-671	71	4	=	=	SYM
cana-671	71	5	𝑏	𝑏	NOUN
cana-671	71	6	adding	add	VERB
cana-671	71	7	(	(	PUNCT
cana-671	71	8	1	1	NUM
cana-671	71	9	2	2	NUM
cana-671	71	10	𝑎	𝑎	NOUN
cana-671	71	11	)	)	PUNCT
cana-671	71	12	2	2	NUM
cana-671	71	13	to	to	ADP
cana-671	71	14	both	both	DET
cana-671	71	15	sides	side	NOUN
cana-671	71	16	and	and	CCONJ
cana-671	71	17	taking	take	VERB
cana-671	71	18	the	the	DET
cana-671	71	19	square	square	ADJ
cana-671	71	20	root	root	NOUN
cana-671	71	21	,	,	PUNCT
cana-671	71	22	thus	thus	ADV
cana-671	71	23	:	:	PUNCT
cana-671	71	24	𝑥	𝑥	X
cana-671	72	1	+	+	NOUN
cana-671	72	2	1	1	NUM
cana-671	72	3	2	2	NUM
cana-671	72	4	𝑎	𝑎	NOUN
cana-671	72	5	=	=	SYM
cana-671	72	6	√	√	PROPN
cana-671	72	7	(	(	PUNCT
cana-671	72	8	1	1	NUM
cana-671	72	9	2	2	NUM
cana-671	72	10	)	)	PUNCT
cana-671	72	11	2	2	NUM
cana-671	73	1	+	+	CCONJ
cana-671	73	2	𝑏	𝑏	PROPN
cana-671	73	3	al	al	PROPN
cana-671	73	4	-	-	PUNCT
cana-671	73	5	khorezmi	khorezmi	PROPN
cana-671	73	6	(	(	PUNCT
cana-671	73	7	1	1	NUM
cana-671	73	8	)	)	PUNCT
cana-671	73	9	presents	present	VERB
cana-671	73	10	two	two	NUM
cana-671	73	11	geometric	geometric	ADJ
cana-671	73	12	demonstrations	demonstration	NOUN
cana-671	73	13	of	of	ADP
cana-671	73	14	identity	identity	NOUN
cana-671	73	15	.	.	PUNCT
cana-671	74	1	figure	figure	NOUN
cana-671	74	2	2	2	NUM
cana-671	74	3	displays	display	VERB
cana-671	74	4	the	the	DET
cana-671	74	5	scheme	scheme	NOUN
cana-671	74	6	that	that	PRON
cana-671	74	7	accompanies	accompany	VERB
cana-671	74	8	his	his	PRON
cana-671	74	9	second	second	ADJ
cana-671	74	10	proof	proof	NOUN
cana-671	74	11	,	,	PUNCT
cana-671	74	12	positioned	position	VERB
cana-671	74	13	next	next	ADV
cana-671	74	14	to	to	ADP
cana-671	74	15	euclid	euclid	PROPN
cana-671	74	16	's	's	PART
cana-671	74	17	schematic	schematic	ADJ
cana-671	74	18	picture	picture	NOUN
cana-671	74	19	of	of	ADP
cana-671	74	20	his	his	PRON
cana-671	74	21	demonstration	demonstration	NOUN
cana-671	74	22	.	.	PUNCT
cana-671	75	1	communications	communication	NOUN
cana-671	75	2	on	on	ADP
cana-671	75	3	applied	apply	VERB
cana-671	75	4	nonlinear	nonlinear	ADJ
cana-671	75	5	analysis	analysis	NOUN
cana-671	75	6	issn	issn	NOUN
cana-671	75	7	:	:	PUNCT
cana-671	75	8	1074	1074	NUM
cana-671	75	9	-	-	PUNCT
cana-671	75	10	133x	133x	NUM
cana-671	75	11	vol	vol	NOUN
cana-671	75	12	31	31	NUM
cana-671	75	13	no	no	NOUN
cana-671	75	14	.	.	PUNCT
cana-671	76	1	2s	2s	NUM
cana-671	76	2	(	(	PUNCT
cana-671	76	3	2024	2024	NUM
cana-671	76	4	)	)	PUNCT
cana-671	76	5	575	575	NUM
cana-671	76	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-671	76	7	figure	figure	NOUN
cana-671	76	8	2	2	NUM
cana-671	76	9	.	.	PUNCT
cana-671	77	1	the	the	DET
cana-671	77	2	text	text	NOUN
cana-671	77	3	mentions	mention	VERB
cana-671	77	4	the	the	DET
cana-671	77	5	schemes	scheme	NOUN
cana-671	77	6	from	from	ADP
cana-671	77	7	al	al	PROPN
cana-671	77	8	-	-	PUNCT
cana-671	77	9	khorezmi	khorezmi	PROPN
cana-671	77	10	's	's	PART
cana-671	77	11	algebra	algebra	NOUN
cana-671	77	12	and	and	CCONJ
cana-671	77	13	euclid	euclid	PROPN
cana-671	77	14	's	's	PART
cana-671	77	15	scheme	scheme	NOUN
cana-671	77	16	.	.	PUNCT
cana-671	78	1	by	by	ADP
cana-671	78	2	comparing	compare	VERB
cana-671	78	3	the	the	DET
cana-671	78	4	babylonian	babylonian	ADJ
cana-671	78	5	texts	text	NOUN
cana-671	78	6	with	with	ADP
cana-671	78	7	euclid	euclid	PROPN
cana-671	78	8	,	,	PUNCT
cana-671	78	9	al	al	PROPN
cana-671	78	10	-	-	PUNCT
cana-671	78	11	quarism	quarism	NOUN
cana-671	78	12	,	,	PUNCT
cana-671	78	13	and	and	CCONJ
cana-671	78	14	omar	omar	PROPN
cana-671	78	15	khayyam	khayyam	PROPN
cana-671	78	16	,	,	PUNCT
cana-671	78	17	we	we	PRON
cana-671	78	18	may	may	AUX
cana-671	78	19	identify	identify	VERB
cana-671	78	20	three	three	NUM
cana-671	78	21	distinct	distinct	ADJ
cana-671	78	22	styles	style	NOUN
cana-671	78	23	of	of	ADP
cana-671	78	24	algebra	algebra	NOUN
cana-671	78	25	.	.	PUNCT
cana-671	79	1	1	1	NUM
cana-671	79	2	.	.	PUNCT
cana-671	79	3	)	)	PUNCT
cana-671	79	4	babylonian	babylonian	ADJ
cana-671	79	5	algebra	algebra	NOUN
cana-671	79	6	,	,	PUNCT
cana-671	79	7	which	which	PRON
cana-671	79	8	combines	combine	VERB
cana-671	79	9	lengths	length	NOUN
cana-671	79	10	and	and	CCONJ
cana-671	79	11	areas	area	NOUN
cana-671	79	12	,	,	PUNCT
cana-671	79	13	adds	add	VERB
cana-671	79	14	them	they	PRON
cana-671	79	15	together	together	ADV
cana-671	79	16	,	,	PUNCT
cana-671	79	17	and	and	CCONJ
cana-671	79	18	equates	equate	VERB
cana-671	79	19	them	they	PRON
cana-671	79	20	with	with	ADP
cana-671	79	21	integers	integer	NOUN
cana-671	79	22	;	;	PUNCT
cana-671	79	23	2	2	NUM
cana-671	79	24	.	.	NUM
cana-671	79	25	)	)	PUNCT
cana-671	79	26	numerical	numerical	PROPN
cana-671	79	27	algebra	algebra	PROPN
cana-671	79	28	,	,	PUNCT
cana-671	79	29	specifically	specifically	ADV
cana-671	79	30	focusing	focus	VERB
cana-671	79	31	on	on	ADP
cana-671	79	32	equations	equation	NOUN
cana-671	79	33	with	with	ADP
cana-671	79	34	rational	rational	ADJ
cana-671	79	35	numbers	number	NOUN
cana-671	79	36	as	as	ADP
cana-671	79	37	coefficients	coefficient	NOUN
cana-671	79	38	and	and	CCONJ
cana-671	79	39	solutions	solution	NOUN
cana-671	79	40	,	,	PUNCT
cana-671	79	41	similar	similar	ADJ
cana-671	79	42	to	to	ADP
cana-671	79	43	diophantus	diophantus	NOUN
cana-671	79	44	'	'	PUNCT
cana-671	79	45	arithmetic	arithmetic	ADJ
cana-671	79	46	;	;	PUNCT
cana-671	79	47	3	3	NUM
cana-671	79	48	.	.	PUNCT
cana-671	79	49	)	)	PUNCT
cana-671	79	50	geometric	geometric	ADJ
cana-671	79	51	algebra	algebra	NOUN
cana-671	79	52	,	,	PUNCT
cana-671	79	53	similar	similar	ADJ
cana-671	79	54	to	to	ADP
cana-671	79	55	omar	omar	PROPN
cana-671	79	56	khayyam	khayyam	PROPN
cana-671	79	57	's	's	PART
cana-671	79	58	algebra	algebra	NOUN
cana-671	79	59	,	,	PUNCT
cana-671	79	60	involves	involve	VERB
cana-671	79	61	the	the	DET
cana-671	79	62	separation	separation	NOUN
cana-671	79	63	of	of	ADP
cana-671	79	64	lengths	length	NOUN
cana-671	79	65	,	,	PUNCT
cana-671	79	66	angles	angle	NOUN
cana-671	79	67	,	,	PUNCT
cana-671	79	68	and	and	CCONJ
cana-671	79	69	volumes	volume	NOUN
cana-671	79	70	;	;	PUNCT
cana-671	79	71	5.multivectors	5.multivectors	NUM
cana-671	79	72	we	we	PRON
cana-671	79	73	can	can	AUX
cana-671	79	74	express	express	VERB
cana-671	79	75	a	a	DET
cana-671	79	76	multivector	multivector	NOUN
cana-671	79	77	as	as	ADP
cana-671	79	78	a	a	DET
cana-671	79	79	combination	combination	NOUN
cana-671	79	80	of	of	ADP
cana-671	79	81	many	many	ADJ
cana-671	79	82	k	k	NOUN
cana-671	79	83	-	-	NOUN
cana-671	79	84	directions	direction	NOUN
cana-671	79	85	,	,	PUNCT
cana-671	79	86	where	where	SCONJ
cana-671	79	87	k	k	PROPN
cana-671	79	88	represents	represent	VERB
cana-671	79	89	distinct	distinct	ADJ
cana-671	79	90	dimensions	dimension	NOUN
cana-671	79	91	or	or	CCONJ
cana-671	79	92	directions	direction	NOUN
cana-671	79	93	.	.	PUNCT
cana-671	80	1	in	in	ADP
cana-671	80	2	the	the	DET
cana-671	80	3	two	two	NUM
cana-671	80	4	-	-	PUNCT
cana-671	80	5	dimensional	dimensional	ADJ
cana-671	80	6	space	space	NOUN
cana-671	80	7	r^2	r^2	NOUN
cana-671	80	8	,	,	PUNCT
cana-671	80	9	it	it	PRON
cana-671	80	10	will	will	AUX
cana-671	80	11	consist	consist	VERB
cana-671	80	12	of	of	ADP
cana-671	80	13	three	three	NUM
cana-671	80	14	components	component	NOUN
cana-671	80	15	:	:	PUNCT
cana-671	80	16	a	a	DET
cana-671	80	17	scalar	scalar	ADJ
cana-671	80	18	component	component	NOUN
cana-671	80	19	,	,	PUNCT
cana-671	80	20	a	a	DET
cana-671	80	21	vector	vector	NOUN
cana-671	80	22	component	component	NOUN
cana-671	80	23	,	,	PUNCT
cana-671	80	24	and	and	CCONJ
cana-671	80	25	a	a	DET
cana-671	80	26	bivector	bivector	NOUN
cana-671	80	27	component	component	NOUN
cana-671	80	28	.	.	PUNCT
cana-671	81	1	𝑎1⏟	𝑎1⏟	PROPN
cana-671	81	2	+	+	CCONJ
cana-671	81	3	(	(	PUNCT
cana-671	81	4	scalar	scalar	ADJ
cana-671	81	5	part	part	NOUN
cana-671	81	6	)	)	PUNCT
cana-671	81	7	𝑎2𝑒1+𝑎3𝑒2⏟	𝑎2𝑒1+𝑎3𝑒2⏟	PROPN
cana-671	81	8	(	(	PUNCT
cana-671	81	9	vector	vector	NOUN
cana-671	81	10	part	part	NOUN
cana-671	81	11	)	)	PUNCT
cana-671	82	1	+	+	CCONJ
cana-671	82	2	𝑎4	𝑎4	PROPN
cana-671	82	3	𝐼⏟	𝐼⏟	ADJ
cana-671	82	4	(	(	PUNCT
cana-671	82	5	bivector	bivector	NOUN
cana-671	82	6	part	part	NOUN
cana-671	82	7	)	)	PUNCT
cana-671	82	8	where	where	SCONJ
cana-671	82	9	a_i	a_i	PROPN
cana-671	82	10	refers	refer	VERB
cana-671	82	11	to	to	ADP
cana-671	82	12	the	the	DET
cana-671	82	13	real	real	ADJ
cana-671	82	14	integers	integer	NOUN
cana-671	82	15	,	,	PUNCT
cana-671	82	16	specifically	specifically	ADV
cana-671	82	17	the	the	DET
cana-671	82	18	components	component	NOUN
cana-671	82	19	of	of	ADP
cana-671	82	20	the	the	DET
cana-671	82	21	multivector	multivector	NOUN
cana-671	82	22	.	.	PUNCT
cana-671	83	1	take	take	VERB
cana-671	83	2	note	note	NOUN
cana-671	83	3	of	of	ADP
cana-671	83	4	the	the	DET
cana-671	83	5	variable	variable	NOUN
cana-671	83	6	a_i	a_i	PROPN
cana-671	83	7	,	,	PUNCT
cana-671	83	8	which	which	PRON
cana-671	83	9	has	have	VERB
cana-671	83	10	the	the	DET
cana-671	83	11	possibility	possibility	NOUN
cana-671	83	12	of	of	ADP
cana-671	83	13	being	be	AUX
cana-671	83	14	zero	zero	NUM
cana-671	83	15	.	.	PUNCT
cana-671	84	1	this	this	PRON
cana-671	84	2	suggests	suggest	VERB
cana-671	84	3	that	that	SCONJ
cana-671	84	4	we	we	PRON
cana-671	84	5	can	can	AUX
cana-671	84	6	also	also	ADV
cana-671	84	7	represent	represent	VERB
cana-671	84	8	the	the	DET
cana-671	84	9	directions	direction	NOUN
cana-671	84	10	as	as	ADP
cana-671	84	11	multivectors	multivector	NOUN
cana-671	84	12	.	.	PUNCT
cana-671	85	1	for	for	ADP
cana-671	85	2	instance	instance	NOUN
cana-671	85	3	,	,	PUNCT
cana-671	85	4	if	if	SCONJ
cana-671	85	5	a_1	a_1	ADJ
cana-671	85	6	and	and	CCONJ
cana-671	85	7	a_4	a_4	ADV
cana-671	85	8	are	be	AUX
cana-671	85	9	both	both	ADV
cana-671	85	10	equal	equal	ADJ
cana-671	85	11	to	to	ADP
cana-671	85	12	zero	zero	NUM
cana-671	85	13	,	,	PUNCT
cana-671	85	14	we	we	PRON
cana-671	85	15	obtain	obtain	VERB
cana-671	85	16	a	a	DET
cana-671	85	17	unidirectional	unidirectional	ADJ
cana-671	85	18	vector	vector	NOUN
cana-671	85	19	.	.	PUNCT
cana-671	86	1	6.bivectors	6.bivectors	NUM
cana-671	86	2	geometric	geometric	ADJ
cana-671	86	3	algebra	algebra	NOUN
cana-671	86	4	introduces	introduce	VERB
cana-671	86	5	an	an	DET
cana-671	86	6	operator	operator	NOUN
cana-671	86	7	that	that	PRON
cana-671	86	8	can	can	AUX
cana-671	86	9	be	be	AUX
cana-671	86	10	considered	consider	VERB
cana-671	86	11	the	the	DET
cana-671	86	12	reciprocal	reciprocal	ADJ
cana-671	86	13	or	or	CCONJ
cana-671	86	14	opposite	opposite	ADJ
cana-671	86	15	of	of	ADP
cana-671	86	16	the	the	DET
cana-671	86	17	scalar	scalar	ADJ
cana-671	86	18	product	product	NOUN
cana-671	86	19	.	.	PUNCT
cana-671	87	1	we	we	PRON
cana-671	87	2	refer	refer	VERB
cana-671	87	3	to	to	ADP
cana-671	87	4	this	this	DET
cana-671	87	5	operation	operation	NOUN
cana-671	87	6	as	as	ADP
cana-671	87	7	the	the	DET
cana-671	87	8	outer	outer	ADJ
cana-671	87	9	product	product	NOUN
cana-671	87	10	,	,	PUNCT
cana-671	87	11	where	where	SCONJ
cana-671	87	12	we	we	PRON
cana-671	87	13	elongate	elongate	VERB
cana-671	87	14	one	one	NUM
cana-671	87	15	vector	vector	NOUN
cana-671	87	16	in	in	ADP
cana-671	87	17	the	the	DET
cana-671	87	18	direction	direction	NOUN
cana-671	87	19	of	of	ADP
cana-671	87	20	another	another	PRON
cana-671	87	21	,	,	PUNCT
cana-671	87	22	instead	instead	ADV
cana-671	87	23	of	of	ADP
cana-671	87	24	projecting	project	VERB
cana-671	87	25	one	one	NUM
cana-671	87	26	vector	vector	NOUN
cana-671	87	27	onto	onto	ADP
cana-671	87	28	another	another	PRON
cana-671	87	29	.	.	PUNCT
cana-671	88	1	to	to	PART
cana-671	88	2	indicate	indicate	VERB
cana-671	88	3	this	this	DET
cana-671	88	4	operator	operator	NOUN
cana-671	88	5	,	,	PUNCT
cana-671	88	6	the	the	DET
cana-671	88	7	symbol	symbol	NOUN
cana-671	88	8	⊥	⊥	X
cana-671	88	9	(	(	PUNCT
cana-671	88	10	and	and	CCONJ
cana-671	88	11	)	)	PUNCT
cana-671	88	12	is	be	AUX
cana-671	88	13	used	use	VERB
cana-671	88	14	.	.	PUNCT
cana-671	89	1	figure	figure	NOUN
cana-671	89	2	3	3	NUM
cana-671	89	3	illustrates	illustrate	VERB
cana-671	89	4	the	the	DET
cana-671	89	5	outer	outer	ADJ
cana-671	89	6	product	product	NOUN
cana-671	89	7	of	of	ADP
cana-671	89	8	two	two	NUM
cana-671	89	9	vectors	vector	NOUN
cana-671	89	10	,	,	PUNCT
cana-671	89	11	symbolized	symbolize	VERB
cana-671	89	12	as	as	ADP
cana-671	89	13	a	a	DET
cana-671	89	14	b.	b.	PROPN
cana-671	89	15	figure	figure	NOUN
cana-671	89	16	3.vector	3.vector	PROPN
cana-671	89	17	an	an	DET
cana-671	89	18	aligns	align	NOUN
cana-671	89	19	with	with	ADP
cana-671	89	20	vector	vector	NOUN
cana-671	89	21	b	b	NOUN
cana-671	89	22	and	and	CCONJ
cana-671	89	23	extends	extend	VERB
cana-671	89	24	in	in	ADP
cana-671	89	25	the	the	DET
cana-671	89	26	same	same	ADJ
cana-671	89	27	direction	direction	NOUN
cana-671	89	28	.	.	PUNCT
cana-671	90	1	communications	communication	NOUN
cana-671	90	2	on	on	ADP
cana-671	90	3	applied	apply	VERB
cana-671	90	4	nonlinear	nonlinear	ADJ
cana-671	90	5	analysis	analysis	NOUN
cana-671	90	6	issn	issn	NOUN
cana-671	90	7	:	:	PUNCT
cana-671	90	8	1074	1074	NUM
cana-671	90	9	-	-	PUNCT
cana-671	90	10	133x	133x	NUM
cana-671	90	11	vol	vol	NOUN
cana-671	90	12	31	31	NUM
cana-671	90	13	no	no	NOUN
cana-671	90	14	.	.	PUNCT
cana-671	91	1	2s	2s	NUM
cana-671	91	2	(	(	PUNCT
cana-671	91	3	2024	2024	NUM
cana-671	91	4	)	)	PUNCT
cana-671	91	5	576	576	NUM
cana-671	91	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-671	91	7	a	a	DET
cana-671	91	8	bivector	bivector	NOUN
cana-671	91	9	is	be	AUX
cana-671	91	10	the	the	DET
cana-671	91	11	term	term	NOUN
cana-671	91	12	we	we	PRON
cana-671	91	13	use	use	VERB
cana-671	91	14	to	to	PART
cana-671	91	15	refer	refer	VERB
cana-671	91	16	to	to	ADP
cana-671	91	17	the	the	DET
cana-671	91	18	resulting	result	VERB
cana-671	91	19	unit	unit	NOUN
cana-671	91	20	,	,	PUNCT
cana-671	91	21	which	which	PRON
cana-671	91	22	is	be	AUX
cana-671	91	23	a	a	DET
cana-671	91	24	2	2	NUM
cana-671	91	25	-	-	PUNCT
cana-671	91	26	dimensional	dimensional	ADJ
cana-671	91	27	subspace	subspace	NOUN
cana-671	91	28	.	.	PUNCT
cana-671	92	1	it	it	PRON
cana-671	92	2	is	be	AUX
cana-671	92	3	important	important	ADJ
cana-671	92	4	to	to	PART
cana-671	92	5	observe	observe	VERB
cana-671	92	6	that	that	SCONJ
cana-671	92	7	a	a	DET
cana-671	92	8	bivector	bivector	NOUN
cana-671	92	9	lacks	lack	VERB
cana-671	92	10	any	any	DET
cana-671	92	11	discernible	discernible	ADJ
cana-671	92	12	form	form	NOUN
cana-671	92	13	.	.	PUNCT
cana-671	93	1	figure	figure	NOUN
cana-671	93	2	4	4	NUM
cana-671	93	3	.	.	PUNCT
cana-671	94	1	the	the	DET
cana-671	94	2	direction	direction	NOUN
cana-671	94	3	of	of	ADP
cana-671	94	4	vector	vector	NOUN
cana-671	94	5	a	a	DET
cana-671	94	6	causes	cause	NOUN
cana-671	94	7	vector	vector	NOUN
cana-671	94	8	b	b	NOUN
cana-671	94	9	to	to	PART
cana-671	94	10	elongate	elongate	VERB
cana-671	94	11	.	.	PUNCT
cana-671	95	1	7.trivectors	7.trivectors	NUM
cana-671	95	2	up	up	ADV
cana-671	95	3	until	until	ADP
cana-671	95	4	this	this	DET
cana-671	95	5	point	point	NOUN
cana-671	95	6	,	,	PUNCT
cana-671	95	7	we	we	PRON
cana-671	95	8	have	have	AUX
cana-671	95	9	used	use	VERB
cana-671	95	10	the	the	DET
cana-671	95	11	outer	outer	ADJ
cana-671	95	12	product	product	NOUN
cana-671	95	13	as	as	ADP
cana-671	95	14	a	a	DET
cana-671	95	15	binary	binary	ADJ
cana-671	95	16	operator	operator	NOUN
cana-671	95	17	on	on	ADP
cana-671	95	18	two	two	NUM
cana-671	95	19	vectors	vector	NOUN
cana-671	95	20	.	.	PUNCT
cana-671	96	1	the	the	DET
cana-671	96	2	outer	outer	ADJ
cana-671	96	3	product	product	NOUN
cana-671	96	4	combines	combine	VERB
cana-671	96	5	two	two	NUM
cana-671	96	6	1	1	NUM
cana-671	96	7	-	-	PUNCT
cana-671	96	8	dimensional	dimensional	ADJ
cana-671	96	9	subspaces	subspace	NOUN
cana-671	96	10	to	to	PART
cana-671	96	11	form	form	VERB
cana-671	96	12	a	a	DET
cana-671	96	13	2	2	NUM
cana-671	96	14	-	-	PUNCT
cana-671	96	15	dimensional	dimensional	ADJ
cana-671	96	16	subspace	subspace	NOUN
cana-671	96	17	.	.	PUNCT
cana-671	97	1	what	what	PRON
cana-671	97	2	would	would	AUX
cana-671	97	3	happen	happen	VERB
cana-671	97	4	if	if	SCONJ
cana-671	97	5	we	we	PRON
cana-671	97	6	were	be	AUX
cana-671	97	7	to	to	PART
cana-671	97	8	elongate	elongate	VERB
cana-671	97	9	a	a	DET
cana-671	97	10	subspace	subspace	NOUN
cana-671	97	11	with	with	ADP
cana-671	97	12	two	two	NUM
cana-671	97	13	dimensions	dimension	NOUN
cana-671	97	14	along	along	ADP
cana-671	97	15	a	a	DET
cana-671	97	16	space	space	NOUN
cana-671	97	17	with	with	ADP
cana-671	97	18	only	only	ADV
cana-671	97	19	one	one	NUM
cana-671	97	20	dimension	dimension	NOUN
cana-671	97	21	?	?	PUNCT
cana-671	98	1	we	we	PRON
cana-671	98	2	can	can	AUX
cana-671	98	3	calculate	calculate	VERB
cana-671	98	4	the	the	DET
cana-671	98	5	result	result	NOUN
cana-671	98	6	of	of	ADP
cana-671	98	7	the	the	DET
cana-671	98	8	vector	vector	NOUN
cana-671	98	9	triple	triple	ADJ
cana-671	98	10	product	product	NOUN
cana-671	98	11	(	(	PUNCT
cana-671	98	12	ab)c	ab)c	NOUN
cana-671	98	13	.	.	NOUN
cana-671	98	14	intuition	intuition	NOUN
cana-671	98	15	predicts	predict	VERB
cana-671	98	16	that	that	SCONJ
cana-671	98	17	this	this	PRON
cana-671	98	18	will	will	AUX
cana-671	98	19	result	result	VERB
cana-671	98	20	in	in	ADP
cana-671	98	21	a	a	DET
cana-671	98	22	three	three	NUM
cana-671	98	23	-	-	PUNCT
cana-671	98	24	dimensional	dimensional	ADJ
cana-671	98	25	subspace	subspace	NOUN
cana-671	98	26	.	.	PUNCT
cana-671	99	1	figure	figure	NOUN
cana-671	99	2	5	5	NUM
cana-671	99	3	demonstrates	demonstrate	VERB
cana-671	99	4	the	the	DET
cana-671	99	5	correctness	correctness	NOUN
cana-671	99	6	of	of	ADP
cana-671	99	7	this	this	DET
cana-671	99	8	expectation	expectation	NOUN
cana-671	99	9	.	.	PUNCT
cana-671	100	1	figure	figure	NOUN
cana-671	100	2	5	5	NUM
cana-671	100	3	.	.	PUNCT
cana-671	101	1	a	a	DET
cana-671	101	2	trivector	trivector	NOUN
cana-671	101	3	is	be	AUX
cana-671	101	4	a	a	DET
cana-671	101	5	mathematical	mathematical	ADJ
cana-671	101	6	object	object	NOUN
cana-671	101	7	that	that	PRON
cana-671	101	8	represents	represent	VERB
cana-671	101	9	a	a	DET
cana-671	101	10	quantity	quantity	NOUN
cana-671	101	11	with	with	ADP
cana-671	101	12	three	three	NUM
cana-671	101	13	components	component	NOUN
cana-671	101	14	.	.	PUNCT
cana-671	102	1	combining	combine	VERB
cana-671	102	2	a	a	DET
cana-671	102	3	bivector	bivector	NOUN
cana-671	102	4	with	with	ADP
cana-671	102	5	a	a	DET
cana-671	102	6	third	third	ADJ
cana-671	102	7	vector	vector	NOUN
cana-671	102	8	creates	create	VERB
cana-671	102	9	a	a	DET
cana-671	102	10	directed	direct	VERB
cana-671	102	11	volume	volume	NOUN
cana-671	102	12	element	element	NOUN
cana-671	102	13	.	.	PUNCT
cana-671	103	1	the	the	DET
cana-671	103	2	term	term	NOUN
cana-671	103	3	trivecto	trivecto	X
cana-671	103	4	refers	refer	VERB
cana-671	103	5	to	to	ADP
cana-671	103	6	this	this	DET
cana-671	103	7	combination	combination	NOUN
cana-671	103	8	.	.	PUNCT
cana-671	104	1	8	8	X
cana-671	104	2	.	.	X
cana-671	104	3	geometric	geometric	ADJ
cana-671	104	4	algebra	algebra	NOUN
cana-671	104	5	applications	application	NOUN
cana-671	104	6	in	in	ADP
cana-671	104	7	physics	physics	NOUN
cana-671	104	8	and	and	CCONJ
cana-671	104	9	mechanics	mechanic	NOUN
cana-671	104	10	in	in	ADP
cana-671	104	11	this	this	DET
cana-671	104	12	context	context	NOUN
cana-671	104	13	,	,	PUNCT
cana-671	104	14	we	we	PRON
cana-671	104	15	introduce	introduce	VERB
cana-671	104	16	the	the	DET
cana-671	104	17	concept	concept	NOUN
cana-671	104	18	of	of	ADP
cana-671	104	19	geometric	geometric	ADJ
cana-671	104	20	algebra	algebra	NOUN
cana-671	104	21	in	in	ADP
cana-671	104	22	a	a	DET
cana-671	104	23	three	three	NUM
cana-671	104	24	-	-	PUNCT
cana-671	104	25	dimensional	dimensional	ADJ
cana-671	104	26	space	space	NOUN
cana-671	104	27	.	.	PUNCT
cana-671	105	1	by	by	ADP
cana-671	105	2	doing	do	VERB
cana-671	105	3	this	this	PRON
cana-671	105	4	,	,	PUNCT
cana-671	105	5	we	we	PRON
cana-671	105	6	will	will	AUX
cana-671	105	7	be	be	AUX
cana-671	105	8	able	able	ADJ
cana-671	105	9	to	to	PART
cana-671	105	10	construct	construct	VERB
cana-671	105	11	a	a	DET
cana-671	105	12	comprehensive	comprehensive	ADJ
cana-671	105	13	representation	representation	NOUN
cana-671	105	14	of	of	ADP
cana-671	105	15	how	how	SCONJ
cana-671	105	16	geometric	geometric	ADJ
cana-671	105	17	algebra	algebra	NOUN
cana-671	105	18	can	can	AUX
cana-671	105	19	effectively	effectively	ADV
cana-671	105	20	address	address	VERB
cana-671	105	21	intriguing	intriguing	ADJ
cana-671	105	22	challenges	challenge	NOUN
cana-671	105	23	in	in	ADP
cana-671	105	24	the	the	DET
cana-671	105	25	fields	field	NOUN
cana-671	105	26	of	of	ADP
cana-671	105	27	physics	physics	NOUN
cana-671	105	28	and	and	CCONJ
cana-671	105	29	mechanics	mechanic	NOUN
cana-671	105	30	.	.	PUNCT
cana-671	106	1	we	we	PRON
cana-671	106	2	will	will	AUX
cana-671	106	3	now	now	ADV
cana-671	106	4	introduce	introduce	VERB
cana-671	106	5	the	the	DET
cana-671	106	6	fundamental	fundamental	ADJ
cana-671	106	7	concept	concept	NOUN
cana-671	106	8	of	of	ADP
cana-671	106	9	a	a	DET
cana-671	106	10	rotor	rotor	NOUN
cana-671	106	11	,	,	PUNCT
cana-671	106	12	which	which	PRON
cana-671	106	13	serves	serve	VERB
cana-671	106	14	as	as	ADP
cana-671	106	15	a	a	DET
cana-671	106	16	powerful	powerful	ADJ
cana-671	106	17	tool	tool	NOUN
cana-671	106	18	for	for	ADP
cana-671	106	19	defining	define	VERB
cana-671	106	20	rotations	rotation	NOUN
cana-671	106	21	.	.	PUNCT
cana-671	107	1	we	we	PRON
cana-671	107	2	will	will	AUX
cana-671	107	3	investigate	investigate	VERB
cana-671	107	4	the	the	DET
cana-671	107	5	relationship	relationship	NOUN
cana-671	107	6	between	between	ADP
cana-671	107	7	rotors	rotor	NOUN
cana-671	107	8	and	and	CCONJ
cana-671	107	9	bivectors	bivector	NOUN
cana-671	107	10	,	,	PUNCT
cana-671	107	11	as	as	ADV
cana-671	107	12	well	well	ADV
cana-671	107	13	as	as	ADP
cana-671	107	14	the	the	DET
cana-671	107	15	specific	specific	ADJ
cana-671	107	16	equations	equation	NOUN
cana-671	107	17	to	to	PART
cana-671	107	18	which	which	PRON
cana-671	107	19	they	they	PRON
cana-671	107	20	adhere	adhere	VERB
cana-671	107	21	.	.	PUNCT
cana-671	108	1	by	by	ADP
cana-671	108	2	effectively	effectively	ADV
cana-671	108	3	handling	handle	VERB
cana-671	108	4	reflections	reflection	NOUN
cana-671	108	5	and	and	CCONJ
cana-671	108	6	rotations	rotation	NOUN
cana-671	108	7	,	,	PUNCT
cana-671	108	8	geometric	geometric	ADJ
cana-671	108	9	algebra	algebra	NOUN
cana-671	108	10	demonstrates	demonstrate	VERB
cana-671	108	11	its	its	PRON
cana-671	108	12	practicality	practicality	NOUN
cana-671	108	13	.	.	PUNCT
cana-671	109	1	in	in	ADP
cana-671	109	2	order	order	NOUN
cana-671	109	3	to	to	PART
cana-671	109	4	determine	determine	VERB
cana-671	109	5	the	the	DET
cana-671	109	6	process	process	NOUN
cana-671	109	7	of	of	ADP
cana-671	109	8	rotating	rotate	VERB
cana-671	109	9	complex	complex	ADJ
cana-671	109	10	numbers	number	NOUN
cana-671	109	11	,	,	PUNCT
cana-671	109	12	we	we	PRON
cana-671	109	13	utilized	utilize	VERB
cana-671	109	14	this	this	DET
cana-671	109	15	technique	technique	NOUN
cana-671	109	16	to	to	PART
cana-671	109	17	derive	derive	VERB
cana-671	109	18	a	a	DET
cana-671	109	19	formula	formula	NOUN
cana-671	109	20	for	for	ADP
cana-671	109	21	rotating	rotate	VERB
cana-671	109	22	vectors	vector	NOUN
cana-671	109	23	in	in	ADP
cana-671	109	24	a	a	DET
cana-671	109	25	two	two	NUM
cana-671	109	26	-	-	PUNCT
cana-671	109	27	dimensional	dimensional	ADJ
cana-671	109	28	space	space	NOUN
cana-671	109	29	.	.	PUNCT
cana-671	110	1	to	to	PART
cana-671	110	2	create	create	VERB
cana-671	110	3	a	a	DET
cana-671	110	4	positive	positive	ADJ
cana-671	110	5	rotation	rotation	NOUN
cana-671	110	6	through	through	ADP
cana-671	110	7	ϕ	ϕ	NOUN
cana-671	110	8	for	for	ADP
cana-671	110	9	a	a	DET
cana-671	110	10	complex	complex	ADJ
cana-671	110	11	integer	integer	NOUN
cana-671	110	12	z	z	PROPN
cana-671	110	13	,	,	PUNCT
cana-671	110	14	one	one	PRON
cana-671	110	15	can	can	AUX
cana-671	110	16	use	use	VERB
cana-671	110	17	the	the	DET
cana-671	110	18	following	follow	VERB
cana-671	110	19	method	method	NOUN
cana-671	110	20	:	:	PUNCT
cana-671	110	21	𝑧	𝑧	PROPN
cana-671	110	22	→	→	PUNCT
cana-671	110	23	𝑒𝐼𝜑𝑧	𝑒𝐼𝜑𝑧	PROPN
cana-671	110	24	in	in	ADP
cana-671	110	25	a	a	DET
cana-671	110	26	conventional	conventional	ADJ
cana-671	110	27	manner	manner	NOUN
cana-671	110	28	,	,	PUNCT
cana-671	110	29	the	the	DET
cana-671	110	30	exponential	exponential	NOUN
cana-671	110	31	of	of	ADP
cana-671	110	32	a	a	DET
cana-671	110	33	multivector	multivector	NOUN
cana-671	110	34	is	be	AUX
cana-671	110	35	defined	define	VERB
cana-671	110	36	using	use	VERB
cana-671	110	37	a	a	DET
cana-671	110	38	power	power	NOUN
cana-671	110	39	series	series	NOUN
cana-671	110	40	.	.	PUNCT
cana-671	111	1	communications	communication	NOUN
cana-671	111	2	on	on	ADP
cana-671	111	3	applied	apply	VERB
cana-671	111	4	nonlinear	nonlinear	ADJ
cana-671	111	5	analysis	analysis	NOUN
cana-671	111	6	issn	issn	NOUN
cana-671	111	7	:	:	PUNCT
cana-671	111	8	1074	1074	NUM
cana-671	111	9	-	-	PUNCT
cana-671	111	10	133x	133x	NUM
cana-671	111	11	vol	vol	NOUN
cana-671	111	12	31	31	NUM
cana-671	111	13	no	no	NOUN
cana-671	111	14	.	.	PUNCT
cana-671	112	1	2s	2s	NUM
cana-671	112	2	(	(	PUNCT
cana-671	112	3	2024	2024	NUM
cana-671	112	4	)	)	PUNCT
cana-671	112	5	577	577	NUM
cana-671	112	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-671	112	7	figure	figure	NOUN
cana-671	112	8	6	6	NUM
cana-671	112	9	.	.	PUNCT
cana-671	112	10	rotations	rotation	NOUN
cana-671	112	11	certain	certain	ADJ
cana-671	112	12	mathematical	mathematical	ADJ
cana-671	112	13	concepts	concept	NOUN
cana-671	112	14	commonly	commonly	ADV
cana-671	112	15	employed	employ	VERB
cana-671	112	16	in	in	ADP
cana-671	112	17	modern	modern	ADJ
cana-671	112	18	times	time	NOUN
cana-671	112	19	owe	owe	VERB
cana-671	112	20	much	much	ADJ
cana-671	112	21	to	to	ADP
cana-671	112	22	the	the	DET
cana-671	112	23	advancements	advancement	NOUN
cana-671	112	24	of	of	ADP
cana-671	112	25	classical	classical	ADJ
cana-671	112	26	mechanics	mechanic	NOUN
cana-671	112	27	.	.	PUNCT
cana-671	113	1	here	here	ADV
cana-671	113	2	's	be	AUX
cana-671	113	3	an	an	DET
cana-671	113	4	example	example	NOUN
cana-671	113	5	of	of	ADP
cana-671	113	6	applying	apply	VERB
cana-671	113	7	geometric	geometric	ADJ
cana-671	113	8	algebra	algebra	NOUN
cana-671	113	9	to	to	ADP
cana-671	113	10	classical	classical	ADJ
cana-671	113	11	mechanics	mechanic	NOUN
cana-671	113	12	:	:	PUNCT
cana-671	113	13	consider	consider	VERB
cana-671	113	14	a	a	DET
cana-671	113	15	particle	particle	NOUN
cana-671	113	16	with	with	ADP
cana-671	113	17	mass	mass	NOUN
cana-671	113	18	m	m	PROPN
cana-671	113	19	that	that	PRON
cana-671	113	20	is	be	AUX
cana-671	113	21	moving	move	VERB
cana-671	113	22	in	in	ADP
cana-671	113	23	three	three	NUM
cana-671	113	24	-	-	PUNCT
cana-671	113	25	dimensional	dimensional	ADJ
cana-671	113	26	space	space	NOUN
cana-671	113	27	.	.	PUNCT
cana-671	114	1	at	at	ADP
cana-671	114	2	any	any	DET
cana-671	114	3	given	give	VERB
cana-671	114	4	time	time	NOUN
cana-671	114	5	,	,	PUNCT
cana-671	114	6	we	we	PRON
cana-671	114	7	may	may	AUX
cana-671	114	8	describe	describe	VERB
cana-671	114	9	the	the	DET
cana-671	114	10	position	position	NOUN
cana-671	114	11	of	of	ADP
cana-671	114	12	the	the	DET
cana-671	114	13	particle	particle	NOUN
cana-671	114	14	as	as	ADP
cana-671	114	15	a	a	DET
cana-671	114	16	vector	vector	NOUN
cana-671	114	17	r	r	NOUN
cana-671	114	18	(	(	PUNCT
cana-671	114	19	or	or	CCONJ
cana-671	114	20	x	x	X
cana-671	114	21	)	)	PUNCT
cana-671	114	22	and	and	CCONJ
cana-671	114	23	its	its	PRON
cana-671	114	24	velocity	velocity	NOUN
cana-671	114	25	as	as	ADP
cana-671	114	26	a	a	DET
cana-671	114	27	vector	vector	NOUN
cana-671	114	28	v.	v.	CCONJ
cana-671	114	29	we	we	PRON
cana-671	114	30	can	can	AUX
cana-671	114	31	denote	denote	VERB
cana-671	114	32	the	the	DET
cana-671	114	33	vector	vector	NOUN
cana-671	114	34	force	force	NOUN
cana-671	114	35	the	the	DET
cana-671	114	36	particle	particle	NOUN
cana-671	114	37	experiences	experience	NOUN
cana-671	114	38	as	as	SCONJ
cana-671	114	39	f.	f.	PROPN
cana-671	114	40	we	we	PRON
cana-671	114	41	can	can	AUX
cana-671	114	42	represent	represent	VERB
cana-671	114	43	the	the	DET
cana-671	114	44	position	position	NOUN
cana-671	114	45	vector	vector	NOUN
cana-671	114	46	r	r	NOUN
cana-671	114	47	as	as	ADP
cana-671	114	48	a	a	DET
cana-671	114	49	bivector	bivector	NOUN
cana-671	114	50	,	,	PUNCT
cana-671	114	51	which	which	PRON
cana-671	114	52	is	be	AUX
cana-671	114	53	the	the	DET
cana-671	114	54	result	result	NOUN
cana-671	114	55	of	of	ADP
cana-671	114	56	multiplying	multiply	VERB
cana-671	114	57	two	two	NUM
cana-671	114	58	vectors	vector	NOUN
cana-671	114	59	,	,	PUNCT
cana-671	114	60	r_1	r_1	PROPN
cana-671	114	61	and	and	CCONJ
cana-671	114	62	r_2	r_2	PROPN
cana-671	114	63	.	.	PUNCT
cana-671	115	1	𝑟	𝑟	X
cana-671	116	1	=	=	SYM
cana-671	116	2	𝑟1	𝑟1	NOUN
cana-671	117	1	𝑥	𝑥	DET
cana-671	117	2	𝑟2	𝑟2	NOUN
cana-671	117	3	three	three	NUM
cana-671	117	4	vectors	vector	NOUN
cana-671	117	5	,	,	PUNCT
cana-671	117	6	v_1	v_1	PROPN
cana-671	117	7	,	,	PUNCT
cana-671	117	8	v_2	v_2	ADJ
cana-671	117	9	,	,	PUNCT
cana-671	117	10	and	and	CCONJ
cana-671	117	11	v_3	v_3	VERB
cana-671	117	12	,	,	PUNCT
cana-671	117	13	can	can	AUX
cana-671	117	14	multiply	multiply	VERB
cana-671	117	15	to	to	PART
cana-671	117	16	create	create	VERB
cana-671	117	17	a	a	DET
cana-671	117	18	trivector	trivector	NOUN
cana-671	117	19	,	,	PUNCT
cana-671	117	20	which	which	PRON
cana-671	117	21	represents	represent	VERB
cana-671	117	22	the	the	DET
cana-671	117	23	velocity	velocity	NOUN
cana-671	117	24	vector	vector	NOUN
cana-671	117	25	v.	v.	ADP
cana-671	117	26	𝑣	𝑣	X
cana-671	117	27	=	=	PUNCT
cana-671	117	28	𝑣1	𝑣1	PROPN
cana-671	117	29	×	×	PROPN
cana-671	117	30	𝑣2	𝑣2	PROPN
cana-671	117	31	×	×	NOUN
cana-671	117	32	𝑣3	𝑣3	ADJ
cana-671	117	33	a	a	DET
cana-671	117	34	rotation	rotation	NOUN
cana-671	117	35	in	in	ADP
cana-671	117	36	classical	classical	ADJ
cana-671	117	37	mechanics	mechanic	NOUN
cana-671	117	38	refers	refer	VERB
cana-671	117	39	to	to	ADP
cana-671	117	40	a	a	DET
cana-671	117	41	transition	transition	NOUN
cana-671	117	42	that	that	PRON
cana-671	117	43	alters	alter	VERB
cana-671	117	44	the	the	DET
cana-671	117	45	orientation	orientation	NOUN
cana-671	117	46	of	of	ADP
cana-671	117	47	an	an	DET
cana-671	117	48	item	item	NOUN
cana-671	117	49	in	in	ADP
cana-671	117	50	threedimensional	threedimensional	ADJ
cana-671	117	51	space	space	NOUN
cana-671	117	52	while	while	SCONJ
cana-671	117	53	keeping	keep	VERB
cana-671	117	54	its	its	PRON
cana-671	117	55	shape	shape	NOUN
cana-671	117	56	and	and	CCONJ
cana-671	117	57	size	size	NOUN
cana-671	117	58	unchanged	unchanged	ADJ
cana-671	117	59	.	.	PUNCT
cana-671	118	1	a	a	DET
cana-671	118	2	rotation	rotation	NOUN
cana-671	118	3	is	be	AUX
cana-671	118	4	the	the	DET
cana-671	118	5	act	act	NOUN
cana-671	118	6	of	of	ADP
cana-671	118	7	turning	turn	VERB
cana-671	118	8	an	an	DET
cana-671	118	9	object	object	NOUN
cana-671	118	10	around	around	ADP
cana-671	118	11	a	a	DET
cana-671	118	12	stationary	stationary	ADJ
cana-671	118	13	axis	axis	NOUN
cana-671	118	14	or	or	CCONJ
cana-671	118	15	point	point	NOUN
cana-671	118	16	.	.	PUNCT
cana-671	119	1	rotation	rotation	NOUN
cana-671	119	2	is	be	AUX
cana-671	119	3	defined	define	VERB
cana-671	119	4	by	by	ADP
cana-671	119	5	two	two	NUM
cana-671	119	6	components	component	NOUN
cana-671	119	7	:	:	PUNCT
cana-671	119	8	the	the	DET
cana-671	119	9	angle	angle	NOUN
cana-671	119	10	and	and	CCONJ
cana-671	119	11	the	the	DET
cana-671	119	12	axis	axis	NOUN
cana-671	119	13	of	of	ADP
cana-671	119	14	rotation	rotation	NOUN
cana-671	119	15	.	.	PUNCT
cana-671	120	1	typically	typically	ADV
cana-671	120	2	,	,	PUNCT
cana-671	120	3	a	a	DET
cana-671	120	4	rotation	rotation	NOUN
cana-671	120	5	can	can	AUX
cana-671	120	6	be	be	AUX
cana-671	120	7	defined	define	VERB
cana-671	120	8	by	by	ADP
cana-671	120	9	two	two	NUM
cana-671	120	10	key	key	ADJ
cana-671	120	11	parameters	parameter	NOUN
cana-671	120	12	:	:	PUNCT
cana-671	120	13	the	the	DET
cana-671	120	14	angle	angle	NOUN
cana-671	120	15	of	of	ADP
cana-671	120	16	rotation	rotation	NOUN
cana-671	120	17	and	and	CCONJ
cana-671	120	18	the	the	DET
cana-671	120	19	x	x	PROPN
cana-671	120	20	,	,	PUNCT
cana-671	120	21	y	y	PROPN
cana-671	120	22	,	,	PUNCT
cana-671	120	23	and	and	CCONJ
cana-671	120	24	z	z	NOUN
cana-671	120	25	coordinates	coordinate	NOUN
cana-671	120	26	of	of	ADP
cana-671	120	27	the	the	DET
cana-671	120	28	rotation	rotation	NOUN
cana-671	120	29	axis	axis	NOUN
cana-671	120	30	.	.	PUNCT
cana-671	121	1	the	the	DET
cana-671	121	2	rotation	rotation	NOUN
cana-671	121	3	of	of	ADP
cana-671	121	4	a	a	DET
cana-671	121	5	solid	solid	ADJ
cana-671	121	6	object	object	NOUN
cana-671	121	7	around	around	ADP
cana-671	121	8	an	an	DET
cana-671	121	9	axis	axis	NOUN
cana-671	121	10	is	be	AUX
cana-671	121	11	one	one	NUM
cana-671	121	12	example	example	NOUN
cana-671	121	13	of	of	ADP
cana-671	121	14	rotational	rotational	ADJ
cana-671	121	15	motion	motion	NOUN
cana-671	121	16	in	in	ADP
cana-671	121	17	mechanics	mechanic	NOUN
cana-671	121	18	that	that	PRON
cana-671	121	19	geometric	geometric	ADJ
cana-671	121	20	algebra	algebra	NOUN
cana-671	121	21	can	can	AUX
cana-671	121	22	clarify	clarify	VERB
cana-671	121	23	.	.	PUNCT
cana-671	122	1	geometric	geometric	ADJ
cana-671	122	2	algebra	algebra	PROPN
cana-671	122	3	offers	offer	VERB
cana-671	122	4	a	a	DET
cana-671	122	5	precise	precise	ADJ
cana-671	122	6	method	method	NOUN
cana-671	122	7	for	for	ADP
cana-671	122	8	describing	describe	VERB
cana-671	122	9	rotations	rotation	NOUN
cana-671	122	10	,	,	PUNCT
cana-671	122	11	which	which	PRON
cana-671	122	12	helps	help	VERB
cana-671	122	13	streamline	streamline	VERB
cana-671	122	14	calculations	calculation	NOUN
cana-671	122	15	and	and	CCONJ
cana-671	122	16	offer	offer	VERB
cana-671	122	17	a	a	DET
cana-671	122	18	deeper	deep	ADJ
cana-671	122	19	understanding	understanding	NOUN
cana-671	122	20	of	of	ADP
cana-671	122	21	the	the	DET
cana-671	122	22	fundamental	fundamental	ADJ
cana-671	122	23	principles	principle	NOUN
cana-671	122	24	of	of	ADP
cana-671	122	25	physics	physics	NOUN
cana-671	122	26	.	.	PUNCT
cana-671	123	1	in	in	ADP
cana-671	123	2	order	order	NOUN
cana-671	123	3	to	to	PART
cana-671	123	4	demonstrate	demonstrate	VERB
cana-671	123	5	this	this	DET
cana-671	123	6	concept	concept	NOUN
cana-671	123	7	,	,	PUNCT
cana-671	123	8	we	we	PRON
cana-671	123	9	will	will	AUX
cana-671	123	10	examine	examine	VERB
cana-671	123	11	the	the	DET
cana-671	123	12	rotational	rotational	ADJ
cana-671	123	13	movement	movement	NOUN
cana-671	123	14	of	of	ADP
cana-671	123	15	a	a	DET
cana-671	123	16	solid	solid	ADJ
cana-671	123	17	object	object	NOUN
cana-671	123	18	around	around	ADP
cana-671	123	19	an	an	DET
cana-671	123	20	axis	axis	NOUN
cana-671	123	21	passing	pass	VERB
cana-671	123	22	through	through	ADP
cana-671	123	23	a	a	DET
cana-671	123	24	specific	specific	ADJ
cana-671	123	25	point	point	NOUN
cana-671	123	26	p	p	NOUN
cana-671	123	27	in	in	ADP
cana-671	123	28	three	three	NUM
cana-671	123	29	-	-	PUNCT
cana-671	123	30	dimensional	dimensional	ADJ
cana-671	123	31	space	space	NOUN
cana-671	123	32	.	.	PUNCT
cana-671	124	1	we	we	PRON
cana-671	124	2	describe	describe	VERB
cana-671	124	3	the	the	DET
cana-671	124	4	rotation	rotation	NOUN
cana-671	124	5	using	use	VERB
cana-671	124	6	a	a	DET
cana-671	124	7	rotor	rotor	NOUN
cana-671	124	8	r.	r.	NOUN
cana-671	124	9	in	in	ADP
cana-671	124	10	geometric	geometric	ADJ
cana-671	124	11	algebra	algebra	NOUN
cana-671	124	12	,	,	PUNCT
cana-671	124	13	a	a	DET
cana-671	124	14	rotor	rotor	NOUN
cana-671	124	15	is	be	AUX
cana-671	124	16	a	a	DET
cana-671	124	17	multivector	multivector	NOUN
cana-671	124	18	that	that	PRON
cana-671	124	19	represents	represent	VERB
cana-671	124	20	a	a	DET
cana-671	124	21	rotation	rotation	NOUN
cana-671	124	22	in	in	ADP
cana-671	124	23	a	a	DET
cana-671	124	24	space	space	NOUN
cana-671	124	25	with	with	ADP
cana-671	124	26	more	more	ADJ
cana-671	124	27	dimensions	dimension	NOUN
cana-671	124	28	.	.	PUNCT
cana-671	125	1	for	for	ADP
cana-671	125	2	instance	instance	NOUN
cana-671	125	3	,	,	PUNCT
cana-671	125	4	a	a	DET
cana-671	125	5	bivector	bivector	NOUN
cana-671	125	6	,	,	PUNCT
cana-671	125	7	a	a	DET
cana-671	125	8	multivector	multivector	NOUN
cana-671	125	9	with	with	ADP
cana-671	125	10	two	two	NUM
cana-671	125	11	dimensions	dimension	NOUN
cana-671	125	12	,	,	PUNCT
cana-671	125	13	might	might	AUX
cana-671	125	14	denote	denote	VERB
cana-671	125	15	a	a	DET
cana-671	125	16	rotor	rotor	NOUN
cana-671	125	17	in	in	ADP
cana-671	125	18	three	three	NUM
cana-671	125	19	-	-	PUNCT
cana-671	125	20	dimensional	dimensional	ADJ
cana-671	125	21	space	space	NOUN
cana-671	125	22	.	.	PUNCT
cana-671	126	1	the	the	DET
cana-671	126	2	geometric	geometric	ADJ
cana-671	126	3	product	product	NOUN
cana-671	126	4	,	,	PUNCT
cana-671	126	5	which	which	PRON
cana-671	126	6	combines	combine	VERB
cana-671	126	7	the	the	DET
cana-671	126	8	scalar	scalar	ADJ
cana-671	126	9	and	and	CCONJ
cana-671	126	10	vector	vector	NOUN
cana-671	126	11	components	component	NOUN
cana-671	126	12	of	of	ADP
cana-671	126	13	two	two	NUM
cana-671	126	14	multivectors	multivector	NOUN
cana-671	126	15	to	to	PART
cana-671	126	16	generate	generate	VERB
cana-671	126	17	a	a	DET
cana-671	126	18	new	new	ADJ
cana-671	126	19	multivector	multivector	NOUN
cana-671	126	20	,	,	PUNCT
cana-671	126	21	explains	explain	VERB
cana-671	126	22	the	the	DET
cana-671	126	23	effect	effect	NOUN
cana-671	126	24	of	of	ADP
cana-671	126	25	a	a	DET
cana-671	126	26	rotor	rotor	NOUN
cana-671	126	27	on	on	ADP
cana-671	126	28	a	a	DET
cana-671	126	29	vector	vector	NOUN
cana-671	126	30	.	.	PUNCT
cana-671	127	1	applying	apply	VERB
cana-671	127	2	a	a	DET
cana-671	127	3	rotor	rotor	NOUN
cana-671	127	4	to	to	ADP
cana-671	127	5	a	a	DET
cana-671	127	6	vector	vector	NOUN
cana-671	127	7	rotates	rotate	VERB
cana-671	127	8	it	it	PRON
cana-671	127	9	,	,	PUNCT
cana-671	127	10	maintaining	maintain	VERB
cana-671	127	11	its	its	PRON
cana-671	127	12	magnitude	magnitude	NOUN
cana-671	127	13	and	and	CCONJ
cana-671	127	14	direction	direction	NOUN
cana-671	127	15	unchanged	unchanged	ADJ
cana-671	127	16	.	.	PUNCT
cana-671	128	1	the	the	DET
cana-671	128	2	geometric	geometric	ADJ
cana-671	128	3	product	product	NOUN
cana-671	128	4	's	's	PART
cana-671	128	5	combination	combination	NOUN
cana-671	128	6	of	of	ADP
cana-671	128	7	rotors	rotor	NOUN
cana-671	128	8	allows	allow	VERB
cana-671	128	9	for	for	SCONJ
cana-671	128	10	intricate	intricate	ADJ
cana-671	128	11	rotational	rotational	ADJ
cana-671	128	12	movements	movement	NOUN
cana-671	128	13	to	to	PART
cana-671	128	14	be	be	AUX
cana-671	128	15	generated	generate	VERB
cana-671	128	16	.	.	PUNCT
cana-671	129	1	the	the	DET
cana-671	129	2	sum	sum	NOUN
cana-671	129	3	of	of	ADP
cana-671	129	4	two	two	NUM
cana-671	129	5	rotors	rotor	NOUN
cana-671	129	6	is	be	AUX
cana-671	129	7	equivalent	equivalent	ADJ
cana-671	129	8	to	to	ADP
cana-671	129	9	a	a	DET
cana-671	129	10	single	single	ADJ
cana-671	129	11	rotor	rotor	NOUN
cana-671	129	12	,	,	PUNCT
cana-671	129	13	which	which	PRON
cana-671	129	14	represents	represent	VERB
cana-671	129	15	the	the	DET
cana-671	129	16	collective	collective	ADJ
cana-671	129	17	communications	communication	NOUN
cana-671	129	18	on	on	ADP
cana-671	129	19	applied	apply	VERB
cana-671	129	20	nonlinear	nonlinear	ADJ
cana-671	129	21	analysis	analysis	NOUN
cana-671	129	22	issn	issn	NOUN
cana-671	129	23	:	:	PUNCT
cana-671	129	24	1074	1074	NUM
cana-671	129	25	-	-	PUNCT
cana-671	129	26	133x	133x	NUM
cana-671	129	27	vol	vol	NOUN
cana-671	129	28	31	31	NUM
cana-671	129	29	no	no	NOUN
cana-671	129	30	.	.	PUNCT
cana-671	130	1	2s	2s	NUM
cana-671	130	2	(	(	PUNCT
cana-671	130	3	2024	2024	NUM
cana-671	130	4	)	)	PUNCT
cana-671	130	5	578	578	NUM
cana-671	130	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-671	130	7	rotation	rotation	NOUN
cana-671	130	8	.	.	PUNCT
cana-671	131	1	in	in	ADP
cana-671	131	2	mechanics	mechanic	NOUN
cana-671	131	3	,	,	PUNCT
cana-671	131	4	a	a	DET
cana-671	131	5	rotor	rotor	NOUN
cana-671	131	6	is	be	AUX
cana-671	131	7	a	a	DET
cana-671	131	8	revolving	revolve	VERB
cana-671	131	9	component	component	NOUN
cana-671	131	10	of	of	ADP
cana-671	131	11	a	a	DET
cana-671	131	12	mechanical	mechanical	ADJ
cana-671	131	13	system	system	NOUN
cana-671	131	14	that	that	PRON
cana-671	131	15	is	be	AUX
cana-671	131	16	present	present	ADJ
cana-671	131	17	in	in	ADP
cana-671	131	18	many	many	ADJ
cana-671	131	19	machines	machine	NOUN
cana-671	131	20	,	,	PUNCT
cana-671	131	21	such	such	ADJ
cana-671	131	22	as	as	ADP
cana-671	131	23	engines	engine	NOUN
cana-671	131	24	,	,	PUNCT
cana-671	131	25	turbines	turbine	NOUN
cana-671	131	26	,	,	PUNCT
cana-671	131	27	electric	electric	ADJ
cana-671	131	28	motor	motor	NOUN
cana-671	131	29	generators	generator	NOUN
cana-671	131	30	,	,	PUNCT
cana-671	131	31	and	and	CCONJ
cana-671	131	32	so	so	ADV
cana-671	131	33	on	on	ADV
cana-671	131	34	.	.	PUNCT
cana-671	132	1	within	within	ADP
cana-671	132	2	a	a	DET
cana-671	132	3	motor	motor	NOUN
cana-671	132	4	,	,	PUNCT
cana-671	132	5	the	the	DET
cana-671	132	6	rotor	rotor	NOUN
cana-671	132	7	serves	serve	VERB
cana-671	132	8	as	as	ADP
cana-671	132	9	the	the	DET
cana-671	132	10	component	component	NOUN
cana-671	132	11	that	that	PRON
cana-671	132	12	transforms	transform	VERB
cana-671	132	13	linear	linear	ADJ
cana-671	132	14	action	action	NOUN
cana-671	132	15	into	into	ADP
cana-671	132	16	rotary	rotary	ADJ
cana-671	132	17	motion	motion	NOUN
cana-671	132	18	,	,	PUNCT
cana-671	132	19	subsequently	subsequently	ADV
cana-671	132	20	transferring	transfer	VERB
cana-671	132	21	it	it	PRON
cana-671	132	22	to	to	ADP
cana-671	132	23	the	the	DET
cana-671	132	24	wheels	wheel	NOUN
cana-671	132	25	.	.	PUNCT
cana-671	133	1	in	in	ADP
cana-671	133	2	a	a	DET
cana-671	133	3	turbine	turbine	NOUN
cana-671	133	4	,	,	PUNCT
cana-671	133	5	a	a	DET
cana-671	133	6	fluid	fluid	NOUN
cana-671	133	7	propels	propel	VERB
cana-671	133	8	the	the	DET
cana-671	133	9	rotating	rotate	VERB
cana-671	133	10	rotor	rotor	NOUN
cana-671	133	11	,	,	PUNCT
cana-671	133	12	which	which	PRON
cana-671	133	13	converts	convert	VERB
cana-671	133	14	the	the	DET
cana-671	133	15	fluid	fluid	NOUN
cana-671	133	16	's	's	PART
cana-671	133	17	energy	energy	NOUN
cana-671	133	18	into	into	ADP
cana-671	133	19	mechanical	mechanical	ADJ
cana-671	133	20	energy	energy	NOUN
cana-671	133	21	.	.	PUNCT
cana-671	134	1	in	in	ADP
cana-671	134	2	contrast	contrast	NOUN
cana-671	134	3	,	,	PUNCT
cana-671	134	4	in	in	ADP
cana-671	134	5	a	a	DET
cana-671	134	6	generator	generator	NOUN
cana-671	134	7	,	,	PUNCT
cana-671	134	8	the	the	DET
cana-671	134	9	rotor	rotor	NOUN
cana-671	134	10	is	be	AUX
cana-671	134	11	the	the	DET
cana-671	134	12	component	component	NOUN
cana-671	134	13	that	that	PRON
cana-671	134	14	rotates	rotate	VERB
cana-671	134	15	and	and	CCONJ
cana-671	134	16	interacts	interact	VERB
cana-671	134	17	with	with	ADP
cana-671	134	18	a	a	DET
cana-671	134	19	magnetic	magnetic	ADJ
cana-671	134	20	field	field	NOUN
cana-671	134	21	in	in	ADP
cana-671	134	22	order	order	NOUN
cana-671	134	23	to	to	PART
cana-671	134	24	generate	generate	VERB
cana-671	134	25	energy	energy	NOUN
cana-671	134	26	.	.	PUNCT
cana-671	135	1	9.recommendations	9.recommendations	NUM
cana-671	135	2	we	we	PRON
cana-671	135	3	have	have	AUX
cana-671	135	4	formulated	formulate	VERB
cana-671	135	5	hypotheses	hypothesis	NOUN
cana-671	135	6	and	and	CCONJ
cana-671	135	7	derived	derive	VERB
cana-671	135	8	answers	answer	NOUN
cana-671	135	9	in	in	ADP
cana-671	135	10	this	this	DET
cana-671	135	11	article	article	NOUN
cana-671	135	12	,	,	PUNCT
cana-671	135	13	all	all	PRON
cana-671	135	14	firmly	firmly	ADV
cana-671	135	15	grounded	ground	VERB
cana-671	135	16	in	in	ADP
cana-671	135	17	scientific	scientific	ADJ
cana-671	135	18	principles	principle	NOUN
cana-671	135	19	,	,	PUNCT
cana-671	135	20	through	through	ADP
cana-671	135	21	extensive	extensive	ADJ
cana-671	135	22	research	research	NOUN
cana-671	135	23	and	and	CCONJ
cana-671	135	24	data	datum	NOUN
cana-671	135	25	collection	collection	NOUN
cana-671	135	26	on	on	ADP
cana-671	135	27	geometric	geometric	ADJ
cana-671	135	28	algebra	algebra	NOUN
cana-671	135	29	and	and	CCONJ
cana-671	135	30	its	its	PRON
cana-671	135	31	use	use	NOUN
cana-671	135	32	in	in	ADP
cana-671	135	33	physics	physics	NOUN
cana-671	135	34	and	and	CCONJ
cana-671	135	35	mechanics	mechanic	NOUN
cana-671	135	36	.	.	PUNCT
cana-671	136	1	our	our	PRON
cana-671	136	2	expectations	expectation	NOUN
cana-671	136	3	for	for	ADP
cana-671	136	4	this	this	DET
cana-671	136	5	study	study	NOUN
cana-671	136	6	and	and	CCONJ
cana-671	136	7	others	other	NOUN
cana-671	136	8	revolve	revolve	VERB
cana-671	136	9	around	around	ADP
cana-671	136	10	the	the	DET
cana-671	136	11	correlation	correlation	NOUN
cana-671	136	12	between	between	ADP
cana-671	136	13	different	different	ADJ
cana-671	136	14	lemmas	lemma	NOUN
cana-671	136	15	and	and	CCONJ
cana-671	136	16	geometric	geometric	ADJ
cana-671	136	17	algebra	algebra	NOUN
cana-671	136	18	.	.	PUNCT
cana-671	137	1	we	we	PRON
cana-671	137	2	have	have	AUX
cana-671	137	3	endeavored	endeavor	VERB
cana-671	137	4	to	to	PART
cana-671	137	5	provide	provide	VERB
cana-671	137	6	a	a	DET
cana-671	137	7	consistent	consistent	ADJ
cana-671	137	8	explanation	explanation	NOUN
cana-671	137	9	and	and	CCONJ
cana-671	137	10	substantiate	substantiate	VERB
cana-671	137	11	it	it	PRON
cana-671	137	12	through	through	ADP
cana-671	137	13	scientific	scientific	ADJ
cana-671	137	14	methodologies	methodology	NOUN
cana-671	137	15	,	,	PUNCT
cana-671	137	16	relying	rely	VERB
cana-671	137	17	on	on	ADP
cana-671	137	18	the	the	DET
cana-671	137	19	facts	fact	NOUN
cana-671	137	20	and	and	CCONJ
cana-671	137	21	information	information	NOUN
cana-671	137	22	we	we	PRON
cana-671	137	23	have	have	AUX
cana-671	137	24	gathered	gather	VERB
cana-671	137	25	.	.	PUNCT
cana-671	138	1	the	the	DET
cana-671	138	2	execution	execution	NOUN
cana-671	138	3	and	and	CCONJ
cana-671	138	4	utilization	utilization	NOUN
cana-671	138	5	of	of	ADP
cana-671	138	6	these	these	DET
cana-671	138	7	concepts	concept	NOUN
cana-671	138	8	still	still	ADV
cana-671	138	9	require	require	VERB
cana-671	138	10	substantial	substantial	ADJ
cana-671	138	11	work	work	NOUN
cana-671	138	12	.	.	PUNCT
cana-671	139	1	while	while	SCONJ
cana-671	139	2	geometric	geometric	ADJ
cana-671	139	3	algebra	algebra	NOUN
cana-671	139	4	may	may	AUX
cana-671	139	5	initially	initially	ADV
cana-671	139	6	appear	appear	VERB
cana-671	139	7	promising	promise	VERB
cana-671	139	8	for	for	ADP
cana-671	139	9	computational	computational	ADJ
cana-671	139	10	geometry	geometry	NOUN
cana-671	139	11	,	,	PUNCT
cana-671	139	12	the	the	DET
cana-671	139	13	translation	translation	NOUN
cana-671	139	14	from	from	ADP
cana-671	139	15	theory	theory	NOUN
cana-671	139	16	to	to	ADP
cana-671	139	17	practical	practical	ADJ
cana-671	139	18	application	application	NOUN
cana-671	139	19	is	be	AUX
cana-671	139	20	more	more	ADV
cana-671	139	21	complex	complex	ADJ
cana-671	139	22	than	than	SCONJ
cana-671	139	23	anticipated	anticipate	VERB
cana-671	139	24	.	.	PUNCT
cana-671	140	1	here	here	ADV
cana-671	140	2	are	be	AUX
cana-671	140	3	some	some	DET
cana-671	140	4	suggestions	suggestion	NOUN
cana-671	140	5	:	:	PUNCT
cana-671	140	6	1	1	X
cana-671	140	7	.	.	X
cana-671	140	8	examine	examine	VERB
cana-671	140	9	algebraic	algebraic	PROPN
cana-671	140	10	formulae	formulae	NOUN
cana-671	140	11	in	in	ADP
cana-671	140	12	physics	physics	NOUN
cana-671	140	13	to	to	PART
cana-671	140	14	determine	determine	VERB
cana-671	140	15	movements	movement	NOUN
cana-671	140	16	and	and	CCONJ
cana-671	140	17	velocities	velocity	NOUN
cana-671	140	18	.	.	PUNCT
cana-671	141	1	2	2	X
cana-671	141	2	.	.	X
cana-671	141	3	utilize	utilize	VERB
cana-671	141	4	vectors	vector	NOUN
cana-671	141	5	and	and	CCONJ
cana-671	141	6	matrices	matrix	NOUN
cana-671	141	7	to	to	PART
cana-671	141	8	depict	depict	VERB
cana-671	141	9	displacements	displacement	NOUN
cana-671	141	10	and	and	CCONJ
cana-671	141	11	forces	force	NOUN
cana-671	141	12	.	.	PUNCT
cana-671	142	1	3	3	X
cana-671	142	2	.	.	X
cana-671	142	3	examine	examine	VERB
cana-671	142	4	the	the	DET
cana-671	142	5	correlations	correlation	NOUN
cana-671	142	6	between	between	ADP
cana-671	142	7	trigonometry	trigonometry	NOUN
cana-671	142	8	and	and	CCONJ
cana-671	142	9	force	force	NOUN
cana-671	142	10	in	in	ADP
cana-671	142	11	mechanics	mechanic	NOUN
cana-671	142	12	.	.	PUNCT
cana-671	143	1	exploring	explore	VERB
cana-671	143	2	these	these	DET
cana-671	143	3	subjects	subject	NOUN
cana-671	143	4	through	through	ADP
cana-671	143	5	practical	practical	ADJ
cana-671	143	6	examples	example	NOUN
cana-671	143	7	will	will	AUX
cana-671	143	8	help	help	VERB
cana-671	143	9	to	to	PART
cana-671	143	10	establish	establish	VERB
cana-671	143	11	links	link	NOUN
cana-671	143	12	between	between	ADP
cana-671	143	13	mathematics	mathematic	NOUN
cana-671	143	14	and	and	CCONJ
cana-671	143	15	other	other	ADJ
cana-671	143	16	scientific	scientific	ADJ
cana-671	143	17	disciplines	discipline	NOUN
cana-671	143	18	.	.	PUNCT
cana-671	144	1	references	reference	NOUN
cana-671	144	2	[	[	X
cana-671	144	3	1	1	NUM
cana-671	144	4	]	]	X
cana-671	144	5	mackenzie	mackenzie	NOUN
cana-671	144	6	,	,	PUNCT
cana-671	144	7	d.	d.	PROPN
cana-671	144	8	(	(	PUNCT
cana-671	144	9	1988	1988	NUM
cana-671	144	10	)	)	PUNCT
cana-671	144	11	.	.	PUNCT
cana-671	145	1	kneževina	kneževina	PROPN
cana-671	145	2	srbija	srbija	PROPN
cana-671	145	3	(	(	PUNCT
cana-671	145	4	1830	1830	NUM
cana-671	145	5	-	-	SYM
cana-671	145	6	1839	1839	NUM
cana-671	145	7	)	)	PUNCT
cana-671	145	8	.	.	PUNCT
cana-671	146	1	by	by	ADP
cana-671	146	2	radoš	radoš	PROPN
cana-671	146	3	ljušić	ljušić	PROPN
cana-671	146	4	.	.	PUNCT
cana-671	147	1	belgrade	belgrade	PROPN
cana-671	147	2	:	:	PUNCT
cana-671	147	3	srpska	srpska	PROPN
cana-671	147	4	akademija	akademija	PROPN
cana-671	147	5	nauka	nauka	PROPN
cana-671	147	6	i	i	PRON
cana-671	147	7	umetnosti	umetnosti	NOUN
cana-671	147	8	,	,	PUNCT
cana-671	147	9	1986	1986	NUM
cana-671	147	10	.	.	PUNCT
cana-671	148	1	508	508	NUM
cana-671	148	2	pp	pp	NOUN
cana-671	148	3	.	.	PUNCT
cana-671	149	1	slavic	slavic	PROPN
cana-671	149	2	review	review	PROPN
cana-671	149	3	,	,	PUNCT
cana-671	149	4	47(2	47(2	PROPN
cana-671	149	5	)	)	PUNCT
cana-671	149	6	,	,	PUNCT
cana-671	149	7	362–363	362–363	NUM
cana-671	149	8	.	.	PUNCT
cana-671	150	1	https://doi.org/10.2307/2498513	https://doi.org/10.2307/2498513	X
cana-671	151	1	[	[	X
cana-671	151	2	2	2	NUM
cana-671	151	3	]	]	PUNCT
cana-671	151	4	ageno	ageno	NOUN
cana-671	151	5	,	,	PUNCT
cana-671	151	6	m.	m.	NOUN
cana-671	151	7	,	,	PUNCT
cana-671	151	8	cortellessa	cortellessa	PROPN
cana-671	151	9	,	,	PUNCT
cana-671	151	10	g.	g.	PROPN
cana-671	151	11	,	,	PUNCT
cana-671	151	12	&	&	CCONJ
cana-671	151	13	querzoli	querzoli	PROPN
cana-671	151	14	,	,	PUNCT
cana-671	151	15	r.	r.	PROPN
cana-671	151	16	(	(	PUNCT
cana-671	151	17	1953	1953	NUM
cana-671	151	18	,	,	PUNCT
cana-671	151	19	march	march	PROPN
cana-671	151	20	)	)	PUNCT
cana-671	151	21	.	.	PUNCT
cana-671	152	1	sezione	sezione	NOUN
cana-671	152	2	d’urto	d’urto	PROPN
cana-671	152	3	totale	totale	PROPN
cana-671	152	4	di	di	PROPN
cana-671	152	5	alcuni	alcuni	PROPN
cana-671	152	6	elementi	elementi	PROPN
cana-671	152	7	per	per	ADP
cana-671	152	8	neutroni	neutroni	PROPN
cana-671	152	9	di	di	PROPN
cana-671	152	10	14	14	NUM
cana-671	152	11	mev	mev	PROPN
cana-671	152	12	.	.	PUNCT
cana-671	153	1	il	il	PROPN
cana-671	153	2	nuovo	nuovo	PROPN
cana-671	153	3	cimento	cimento	PROPN
cana-671	153	4	,	,	PUNCT
cana-671	153	5	10(3	10(3	NUM
cana-671	153	6	)	)	PUNCT
cana-671	153	7	,	,	PUNCT
cana-671	153	8	281–299	281–299	NUM
cana-671	153	9	.	.	PUNCT
cana-671	154	1	https://doi.org/10.1007/bf02786199	https://doi.org/10.1007/bf02786199	X
cana-671	155	1	[	[	X
cana-671	155	2	3	3	NUM
cana-671	155	3	]	]	X
cana-671	155	4	amin	amin	NOUN
cana-671	155	5	,	,	PUNCT
cana-671	155	6	i.	i.	PROPN
cana-671	155	7	,	,	PUNCT
cana-671	155	8	ibrahim	ibrahim	PROPN
cana-671	155	9	,	,	PUNCT
cana-671	155	10	y.	y.	PROPN
cana-671	155	11	,	,	PUNCT
cana-671	155	12	&	&	CCONJ
cana-671	155	13	yousif	yousif	PROPN
cana-671	155	14	,	,	PUNCT
cana-671	155	15	m.	m.	NOUN
cana-671	155	16	(	(	PUNCT
cana-671	155	17	2013	2013	NUM
cana-671	155	18	,	,	PUNCT
cana-671	155	19	may	may	AUX
cana-671	155	20	21	21	NUM
cana-671	155	21	)	)	PUNCT
cana-671	155	22	.	.	PUNCT
cana-671	156	1	rad	rad	NOUN
cana-671	156	2	-	-	NOUN
cana-671	156	3	projective	projective	ADJ
cana-671	156	4	and	and	CCONJ
cana-671	156	5	strongly	strongly	ADV
cana-671	156	6	rad	rad	ADJ
cana-671	156	7	-	-	PUNCT
cana-671	156	8	projective	projective	ADJ
cana-671	156	9	modules	module	NOUN
cana-671	156	10	.	.	PUNCT
cana-671	157	1	communications	communication	NOUN
cana-671	157	2	in	in	ADP
cana-671	157	3	algebra	algebra	NOUN
cana-671	157	4	,	,	PUNCT
cana-671	157	5	41(6	41(6	NOUN
cana-671	157	6	)	)	PUNCT
cana-671	157	7	,	,	PUNCT
cana-671	157	8	2174–2192	2174–2192	NUM
cana-671	157	9	.	.	PUNCT
cana-671	157	10	https://doi.org/10.1080/00927872.2012.654552	https://doi.org/10.1080/00927872.2012.654552	PUNCT
cana-671	158	1	[	[	X
cana-671	158	2	4	4	NUM
cana-671	158	3	]	]	X
cana-671	158	4	popović	popović	NOUN
cana-671	158	5	-	-	PUNCT
cana-671	158	6	šević	šević	PROPN
cana-671	158	7	,	,	PUNCT
cana-671	158	8	n.	n.	NOUN
cana-671	158	9	(	(	PUNCT
cana-671	158	10	2021	2021	NUM
cana-671	158	11	)	)	PUNCT
cana-671	158	12	.	.	PUNCT
cana-671	159	1	nematerijalno	nematerijalno	PROPN
cana-671	159	2	kulturno	kulturno	PROPN
cana-671	159	3	nasleđe	nasleđe	PROPN
cana-671	159	4	izazovi	izazovi	PROPN
cana-671	159	5	i	i	PROPN
cana-671	159	6	tendencije	tendencije	PROPN
cana-671	159	7	jelena	jelena	PROPN
cana-671	159	8	ćuković	ćuković	PROPN
cana-671	159	9	:	:	PUNCT
cana-671	160	1	nematerijalno	nematerijalno	PROPN
cana-671	160	2	kulturno	kulturno	PROPN
cana-671	160	3	nasleđe	nasleđe	ADV
cana-671	160	4	iz	iz	ADP
cana-671	160	5	antropološke	antropološke	PROPN
cana-671	160	6	perspektive	perspektive	NOUN
cana-671	160	7	,	,	PUNCT
cana-671	160	8	univerzitet	univerzitet	NOUN
cana-671	160	9	u	u	NOUN
cana-671	160	10	beogradu	beogradu	NOUN
cana-671	160	11	,	,	PUNCT
cana-671	160	12	filozofski	filozofski	ADJ
cana-671	160	13	fakultet	fakultet	NOUN
cana-671	160	14	,	,	PUNCT
cana-671	160	15	beograd	beograd	PROPN
cana-671	160	16	,	,	PUNCT
cana-671	160	17	2019	2019	NUM
cana-671	160	18	.	.	PUNCT
cana-671	161	1	zbornik	zbornik	PROPN
cana-671	161	2	radova	radova	PROPN
cana-671	161	3	filozofskog	filozofskog	PROPN
cana-671	161	4	fakulteta	fakulteta	PROPN
cana-671	161	5	u	u	PROPN
cana-671	161	6	pristini	pristini	NOUN
cana-671	161	7	,	,	PUNCT
cana-671	161	8	51(2	51(2	NUM
cana-671	161	9	)	)	PUNCT
cana-671	161	10	,	,	PUNCT
cana-671	161	11	423–429	423–429	NUM
cana-671	161	12	.	.	PUNCT
cana-671	162	1	https://doi.org/10.5937/zrffp51-31493	https://doi.org/10.5937/zrffp51-31493	VERB
cana-671	163	1	[	[	X
cana-671	163	2	5	5	NUM
cana-671	163	3	]	]	PUNCT
cana-671	163	4	choi	choi	NOUN
cana-671	163	5	,	,	PUNCT
cana-671	163	6	s.	s.	PROPN
cana-671	163	7	h.	h.	PROPN
cana-671	163	8	,	,	PUNCT
cana-671	163	9	lee	lee	PROPN
cana-671	163	10	,	,	PUNCT
cana-671	163	11	j.	j.	PROPN
cana-671	163	12	,	,	PUNCT
cana-671	163	13	&	&	CCONJ
cana-671	163	14	nam	nam	PROPN
cana-671	163	15	,	,	PUNCT
cana-671	163	16	k.	k.	PROPN
cana-671	163	17	b.	b.	PROPN
cana-671	163	18	(	(	PUNCT
cana-671	163	19	2009	2009	NUM
cana-671	163	20	,	,	PUNCT
cana-671	163	21	december	december	PROPN
cana-671	163	22	)	)	PUNCT
cana-671	163	23	.	.	PUNCT
cana-671	164	1	algebra	algebra	NOUN
cana-671	164	2	versus	versus	ADP
cana-671	164	3	its	its	PRON
cana-671	164	4	anti	anti	ADJ
cana-671	164	5	-	-	ADJ
cana-671	164	6	symmetric	symmetric	ADJ
cana-671	164	7	algebra	algebra	NOUN
cana-671	164	8	.	.	PUNCT
cana-671	165	1	algebra	algebra	NOUN
cana-671	165	2	colloquium	colloquium	NOUN
cana-671	165	3	,	,	PUNCT
cana-671	165	4	16(04	16(04	NUM
cana-671	165	5	)	)	PUNCT
cana-671	165	6	,	,	PUNCT
cana-671	165	7	661–668	661–668	NUM
cana-671	165	8	.	.	PUNCT
cana-671	166	1	https://doi.org/10.1142/s1005386709000625	https://doi.org/10.1142/s1005386709000625	PRON
cana-671	166	2	[	[	X
cana-671	166	3	6	6	NUM
cana-671	166	4	]	]	X
cana-671	166	5	hitchin	hitchin	X
cana-671	166	6	,	,	PUNCT
cana-671	166	7	n.	n.	PROPN
cana-671	166	8	j.	j.	PROPN
cana-671	166	9	(	(	PUNCT
cana-671	166	10	1985	1985	NUM
cana-671	166	11	,	,	PUNCT
cana-671	166	12	may	may	AUX
cana-671	166	13	)	)	PUNCT
cana-671	166	14	.	.	PUNCT
cana-671	167	1	314	314	X
cana-671	167	2	.	.	PUNCT
cana-671	167	3	dfl.150.-	dfl.150.-	PROPN
cana-671	167	4	;	;	PUNCT
cana-671	167	5	us$58.00	us$58.00	PROPN
cana-671	167	6	.	.	PUNCT
cana-671	168	1	(	(	PUNCT
cana-671	168	2	d.	d.	PROPN
cana-671	168	3	reidel	reidel	PROPN
cana-671	168	4	publishing	publishing	PROPN
cana-671	168	5	company	company	NOUN
cana-671	168	6	,	,	PUNCT
cana-671	168	7	holland	holland	PROPN
cana-671	168	8	,	,	PUNCT
cana-671	168	9	1984	1984	NUM
cana-671	168	10	.	.	PUNCT
cana-671	168	11	)	)	PUNCT
cana-671	168	12	.	.	PUNCT
cana-671	169	1	bulletin	bulletin	NOUN
cana-671	169	2	of	of	ADP
cana-671	169	3	the	the	DET
cana-671	169	4	london	london	PROPN
cana-671	169	5	mathematical	mathematical	ADJ
cana-671	169	6	society	society	NOUN
cana-671	169	7	,	,	PUNCT
cana-671	169	8	17(3	17(3	NUM
cana-671	169	9	)	)	PUNCT
cana-671	169	10	,	,	PUNCT
cana-671	169	11	289–290	289–290	NUM
cana-671	169	12	.	.	PUNCT
cana-671	170	1	https://doi.org/10.1112/blms/17.3.289	https://doi.org/10.1112/blms/17.3.289	PUNCT
cana-671	171	1	[	[	X
cana-671	171	2	7	7	NUM
cana-671	171	3	]	]	SYM
cana-671	171	4	hestenes	hestene	NOUN
cana-671	171	5	,	,	PUNCT
cana-671	171	6	d.	d.	PROPN
cana-671	171	7	,	,	PUNCT
cana-671	171	8	sobczyk	sobczyk	NOUN
cana-671	171	9	,	,	PUNCT
cana-671	171	10	g.	g.	PROPN
cana-671	171	11	,	,	PUNCT
cana-671	171	12	&	&	CCONJ
cana-671	171	13	marsh	marsh	PROPN
cana-671	171	14	,	,	PUNCT
cana-671	171	15	j.	j.	PROPN
cana-671	171	16	s.	s.	PROPN
cana-671	171	17	(	(	PUNCT
cana-671	171	18	1985	1985	NUM
cana-671	171	19	,	,	PUNCT
cana-671	171	20	may	may	AUX
cana-671	171	21	1	1	NUM
cana-671	171	22	)	)	PUNCT
cana-671	171	23	.	.	PUNCT
cana-671	172	1	clifford	clifford	PROPN
cana-671	172	2	algebra	algebra	PROPN
cana-671	172	3	to	to	ADP
cana-671	172	4	geometric	geometric	ADJ
cana-671	172	5	calculus	calculus	NOUN
cana-671	172	6	..	..	PUNCT
cana-671	172	7	american	american	PROPN
cana-671	172	8	journal	journal	PROPN
cana-671	172	9	of	of	ADP
cana-671	172	10	physics	physics	PROPN
cana-671	172	11	,	,	PUNCT
cana-671	172	12	53(5	53(5	NUM
cana-671	172	13	)	)	PUNCT
cana-671	172	14	,	,	PUNCT
cana-671	172	15	510–511	510–511	NUM
cana-671	172	16	.	.	PUNCT
cana-671	172	17	https://doi.org/10.1119/1.14223	https://doi.org/10.1119/1.14223	PROPN
cana-671	173	1	[	[	X
cana-671	173	2	8	8	NUM
cana-671	173	3	]	]	X
cana-671	173	4	helmstetter	helmstetter	NOUN
cana-671	173	5	,	,	PUNCT
cana-671	173	6	j.	j.	PROPN
cana-671	173	7	(	(	PUNCT
cana-671	173	8	1987	1987	NUM
cana-671	173	9	,	,	PUNCT
cana-671	173	10	november	november	PROPN
cana-671	173	11	)	)	PUNCT
cana-671	173	12	.	.	PUNCT
cana-671	174	1	monoïdes	monoïdes	PROPN
cana-671	174	2	de	de	PROPN
cana-671	174	3	clifford	clifford	PROPN
cana-671	174	4	et	et	PROPN
cana-671	174	5	déformations	déformation	VERB
cana-671	174	6	d’algèbres	d’algèbre	NOUN
cana-671	174	7	de	de	ADP
cana-671	174	8	clifford	clifford	PROPN
cana-671	174	9	.	.	PROPN
cana-671	175	1	journal	journal	PROPN
cana-671	175	2	of	of	ADP
cana-671	175	3	algebra	algebra	PROPN
cana-671	175	4	,	,	PUNCT
cana-671	175	5	111(1	111(1	NUM
cana-671	175	6	)	)	PUNCT
cana-671	175	7	,	,	PUNCT
cana-671	175	8	14–48	14–48	NUM
cana-671	175	9	.	.	PUNCT
cana-671	176	1	https://doi.org/10.1016/0021-8693(87)90241-9	https://doi.org/10.1016/0021-8693(87)90241-9	PROPN
cana-671	176	2	[	[	X
cana-671	176	3	9	9	NUM
cana-671	176	4	]	]	PUNCT
cana-671	176	5	lasenby	lasenby	NOUN
cana-671	176	6	,	,	PUNCT
cana-671	176	7	a.	a.	NOUN
cana-671	176	8	n.	n.	NOUN
cana-671	176	9	(	(	PUNCT
cana-671	176	10	2020	2020	NUM
cana-671	176	11	,	,	PUNCT
cana-671	176	12	february	february	PROPN
cana-671	176	13	22	22	NUM
cana-671	176	14	)	)	PUNCT
cana-671	176	15	.	.	PUNCT
cana-671	177	1	a	a	DET
cana-671	177	2	1d	1d	NUM
cana-671	177	3	up	up	ADP
cana-671	177	4	approach	approach	NOUN
cana-671	177	5	to	to	ADP
cana-671	177	6	conformal	conformal	ADJ
cana-671	177	7	geometric	geometric	ADJ
cana-671	177	8	algebra	algebra	NOUN
cana-671	177	9	:	:	PUNCT
cana-671	177	10	applications	application	NOUN
cana-671	177	11	in	in	ADP
cana-671	177	12	line	line	NOUN
cana-671	177	13	fitting	fitting	ADJ
cana-671	177	14	and	and	CCONJ
cana-671	177	15	quantum	quantum	NOUN
cana-671	177	16	mechanics	mechanic	NOUN
cana-671	177	17	.	.	PUNCT
cana-671	177	18	,	,	PUNCT
cana-671	177	19	30(2	30(2	NUM
cana-671	177	20	)	)	PUNCT
cana-671	177	21	.	.	PUNCT
cana-671	178	1	https://doi.org/10.1007/s00006-020-1046-0	https://doi.org/10.1007/s00006-020-1046-0	NUM
cana-671	178	2	communications	communication	NOUN
cana-671	178	3	on	on	ADP
cana-671	178	4	applied	apply	VERB
cana-671	178	5	nonlinear	nonlinear	ADJ
cana-671	178	6	analysis	analysis	NOUN
cana-671	178	7	issn	issn	NOUN
cana-671	178	8	:	:	PUNCT
cana-671	178	9	1074	1074	NUM
cana-671	178	10	-	-	PUNCT
cana-671	178	11	133x	133x	NUM
cana-671	178	12	vol	vol	NOUN
cana-671	178	13	31	31	NUM
cana-671	178	14	no	no	NOUN
cana-671	178	15	.	.	PUNCT
cana-671	179	1	2s	2s	NUM
cana-671	179	2	(	(	PUNCT
cana-671	179	3	2024	2024	NUM
cana-671	179	4	)	)	PUNCT
cana-671	179	5	579	579	NUM
cana-671	179	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-671	180	1	[	[	X
cana-671	180	2	10	10	NUM
cana-671	180	3	]	]	PUNCT
cana-671	180	4	balbes	balbe	NOUN
cana-671	180	5	,	,	PUNCT
cana-671	180	6	r.	r.	PROPN
cana-671	180	7	(	(	PUNCT
cana-671	180	8	1996	1996	NUM
cana-671	180	9	,	,	PUNCT
cana-671	180	10	september	september	PROPN
cana-671	180	11	)	)	PUNCT
cana-671	180	12	.	.	PUNCT
cana-671	181	1	a	a	DET
cana-671	181	2	lattice	lattice	NOUN
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cana-671	181	4	interpolants	interpolant	NOUN
cana-671	181	5	with	with	ADP
cana-671	181	6	applications	application	NOUN
cana-671	181	7	to	to	ADP
cana-671	181	8	computer	computer	NOUN
cana-671	181	9	graphics	graphic	NOUN
cana-671	181	10	.	.	PUNCT
cana-671	182	1	algebra	algebra	PROPN
cana-671	182	2	universalis	universali	VERB
cana-671	182	3	,	,	PUNCT
cana-671	182	4	36(3	36(3	NOUN
cana-671	182	5	)	)	PUNCT
cana-671	182	6	,	,	PUNCT
cana-671	182	7	331–356	331–356	NUM
cana-671	182	8	.	.	PUNCT
cana-671	183	1	https://doi.org/10.1007/bf01236760	https://doi.org/10.1007/bf01236760	X
cana-671	184	1	[	[	X
cana-671	184	2	11	11	NUM
cana-671	184	3	]	]	X
cana-671	184	4	sokhatsky	sokhatsky	ADJ
cana-671	184	5	,	,	PUNCT
cana-671	184	6	f.	f.	PROPN
cana-671	184	7	m.	m.	PROPN
cana-671	184	8	(	(	PUNCT
cana-671	184	9	2004	2004	NUM
cana-671	184	10	)	)	PUNCT
cana-671	184	11	.	.	PUNCT
cana-671	185	1	commutation	commutation	NOUN
cana-671	185	2	of	of	ADP
cana-671	185	3	operations	operation	NOUN
cana-671	185	4	and	and	CCONJ
cana-671	185	5	its	its	PRON
cana-671	185	6	relationship	relationship	NOUN
cana-671	185	7	with	with	ADP
cana-671	185	8	menger	menger	PROPN
cana-671	185	9	and	and	CCONJ
cana-671	185	10	mann	mann	PROPN
cana-671	185	11	superpositions	superpositions	PROPN
cana-671	185	12	.	.	PUNCT
cana-671	186	1	discussiones	discussione	NOUN
cana-671	186	2	mathematicae	mathematicae	VERB
cana-671	186	3	general	general	ADJ
cana-671	186	4	algebra	algebra	PROPN
cana-671	186	5	and	and	CCONJ
cana-671	186	6	applications	application	NOUN
cana-671	186	7	,	,	PUNCT
cana-671	186	8	24(2	24(2	NUM
cana-671	186	9	)	)	PUNCT
cana-671	186	10	,	,	PUNCT
cana-671	186	11	153	153	NUM
cana-671	186	12	.	.	PUNCT
cana-671	186	13	https://doi.org/10.7151/dmgaa.1082	https://doi.org/10.7151/dmgaa.1082	PROPN
cana-671	187	1	[	[	X
cana-671	187	2	12	12	NUM
cana-671	187	3	]	]	X
cana-671	187	4	hrdina	hrdina	PROPN
cana-671	187	5	,	,	PUNCT
cana-671	187	6	j.	j.	PROPN
cana-671	187	7	,	,	PUNCT
cana-671	187	8	návrat	návrat	PROPN
cana-671	187	9	,	,	PUNCT
cana-671	187	10	a.	a.	NOUN
cana-671	187	11	,	,	PUNCT
cana-671	187	12	vašík	vašík	NOUN
cana-671	187	13	,	,	PUNCT
cana-671	187	14	p.	p.	NOUN
cana-671	187	15	,	,	PUNCT
cana-671	187	16	&	&	CCONJ
cana-671	187	17	dorst	dorst	PROPN
cana-671	187	18	,	,	PUNCT
cana-671	187	19	l.	l.	PROPN
cana-671	187	20	(	(	PUNCT
cana-671	187	21	2021	2021	NUM
cana-671	187	22	,	,	PUNCT
cana-671	187	23	february	february	PROPN
cana-671	187	24	22	22	NUM
cana-671	187	25	)	)	PUNCT
cana-671	187	26	.	.	PUNCT
cana-671	188	1	projective	projective	ADJ
cana-671	188	2	geometric	geometric	ADJ
cana-671	188	3	algebra	algebra	PROPN
cana-671	188	4	as	as	ADP
cana-671	188	5	a	a	DET
cana-671	188	6	subalgebra	subalgebra	NOUN
cana-671	188	7	of	of	ADP
cana-671	188	8	conformal	conformal	ADJ
cana-671	188	9	geometric	geometric	ADJ
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cana-671	188	11	.	.	PUNCT
cana-671	189	1	algebras	algebras	PROPN
cana-671	189	2	,	,	PUNCT
cana-671	189	3	31(2	31(2	NUM
cana-671	189	4	)	)	PUNCT
cana-671	189	5	.	.	PUNCT
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cana-671	191	1	[	[	X
cana-671	191	2	13	13	NUM
cana-671	191	3	]	]	SYM
cana-671	191	4	hansen	hansen	PROPN
cana-671	191	5	,	,	PUNCT
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cana-671	195	5	,	,	PUNCT
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cana-671	195	9	,	,	PUNCT
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cana-671	195	13	.	.	PUNCT
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cana-671	196	5	der	der	PROPN
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cana-671	196	8	.	.	PUNCT
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cana-671	197	7	.	.	PUNCT
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cana-671	199	5	,	,	PUNCT
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cana-671	199	7	,	,	PUNCT
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cana-671	199	11	,	,	PUNCT
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cana-671	199	13	,	,	PUNCT
cana-671	199	14	s.	s.	PROPN
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