Chapter 1 Topics in Analytic Geometry

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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 34Vectors in Coordinate SystemsIf a vector v is positioned with its initial point at the origin of the rectangular coordinatesystem, then the terminal point will have coordinates of the form (v 1 ,v 2 ) or (v 1 ,v 2 ,v 3 ),depending on whether the vector is in 2-space or 3-space. We call these coordinates thecomponents of v, and we write v in component form using the bracket notationv = 〈v 1 ,v 2 〉 or v = 〈v 1 ,v 2 ,v 3 〉In particular, the zero vectors in 2-space and 3-space arerespectively.v = 〈0,0〉 and v = 〈0,0,0〉yz• (v 1 ,v 2 )• (v 1 ,v 2 ,v 3 )vvxyTheorem 3.3 Two vectors are equivalent if and only if their corresponding components areequal.xFor example,〈a,b,c〉 = 〈−2,3,5〉if and only if a = −2, b = 3, and c = 5.Arithmetic Operations on VectorsTheorem 3.4 If v = 〈v 1 ,v 2 〉 and w = 〈w 1 ,w 2 〉 are vectors in 2-space and k is any scalar,thenv+w = 〈v 1 +w 1 ,v 2 +w 2 〉v−w = 〈v 1 −w 1 ,v 2 −w 2 〉kv = 〈kv 1 ,kv 2 〉Similarly, if v = 〈v 1 ,v 2 ,v 3 〉 and w = 〈w 1 ,w 2 ,w 3 〉 are vectors in 3-space and k is any scalar,thenv+w = 〈v 1 +w 1 ,v 2 +w 2 ,v 3 +w 3 〉v−w = 〈v 1 −w 1 ,v 2 −w 2 ,v 3 −w 3 〉kv = 〈kv 1 ,kv 2 ,kv 3 〉
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 35Example 3.5 If v = 〈−1,2,−3〉 and w = 〈2,0,−4〉, thenv+w =3v =−2w =w−3v =Vectors With Initial Point Not At The OriginSuppose that P 1 (x 1 ,y 1 ) and P 2 (x 2 ,y 2 ) are points in 2-space and we interested in findingthe components of the vector −−→ P 1 P 2 . As illustrated in the following figure, we can write thisvector as−−→P 1 P 2 = −−→ OP 2 − −−→ OP 1 = 〈x 2 ,y 2 〉−〈x 1 ,y 1 〉 = 〈x 2 −x 1 ,y 2 −y 1 〉yP 1 (x 1 ,y 1 )•−−→OP 1−−−→P 1 P 2−−→ OP2P 2 (x 2 ,y 2 )•OThus, wehaveshownthatthecomponentsofthevector −−→ P 1 P 2 canbeobtainedbysubtractingthe coordinates of its initial point from the coordinates of its terminal point. Similarcomputations hold in 3-space, so we have established the following result.Theorem 3.5 If −−→ P 1 P 2 is a vector in 2-space with initial point P 1 (x 1 ,y 1 ) and terminal pointP 2 (x 2 ,y 2 ), then−−→P 1 P 2 = 〈x 2 −x 1 ,y 2 −y 1 〉Similarly, if −−→ P 1 P 2 is a vector in 3-space with initial point P 1 (x 1 ,y 1 ,z 1 ) and terminal pointP 2 (x 2 ,y 2 ,z 2 ), then−−→P 1 P 2 = 〈x 2 −x 1 ,y 2 −y 1 ,z 2 −z 1 〉Example 3.6 In 2-space the vector from P 1 (3,2) to P 2 (−1,4) is−−→P 1 P 2 = 〈−1−3,4−2〉 = 〈−4,2〉xand in 3-space the vector from A(1,−2,0) to B(−3,1,2) is−→AB = 〈1−(−3),−2−1,0−2〉 = 〈4,−3,−2〉✠
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Chapter 1 Topics in Analytic Geometry

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