Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 36yP 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〉and in 3-space the vector from A(1,−2,0) to B(−3,1,2) isx−→AB = 〈1−(−3),−2−1,0−2〉 = 〈4,−3,−2〉✠Rules of Vector ArithmeticTheorem 3.6 For any vectors u, v, and w and any scalars k and l, the following relationshipshold:(a) u+v = v+u(b) (u+v)+w = u+(v+w)(c) u+0 = 0+u = u(d) u+(−u) = 0(e) k(lu) = (kl)u(f) k(u+v) = ku+kv(g) (k +l)u = ku+lu(h) 1u = uNorm of a VectorThe distance between the initial and terminal points of a vector v is called the length, thenorm, or the magnitude of v and is denoted by ‖v‖. This distance does not change if