Chapter 1 Topics in Analytic Geometry

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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
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 37the vector is translated. The norm of a vector v = 〈v 1 ,v 2 〉 in 2-space is given by‖v‖ =√v 2 1 +v 2 2and the norm of a vector v = 〈v 1 ,v 2 ,v 3 〉 in 3-space is given by‖v‖ =√v 2 1 +v 2 2 +v 2 3Example 3.7 Find the norm of v = 〈4,−2〉, and w = 〈−1,3,5〉.Solution .........is,For any vector v and scalar k, the length of kv must be |k| times the length of v; that‖kv‖ = |k|‖v‖Thus, for example,‖5v‖ = |5|‖v‖ = 5‖v‖‖−3v‖ = |−3|‖v‖ = 3‖v‖‖−v‖ = |−1|‖v‖ = ‖v‖This applies to vectors in 2-space and 3-space.Unit VectorsA vector of length 1 is called a unit vector. In an xy-coordinate system the unit vectorsalong the x-axis and y-axis are denoted by i and j, respectively; and in xyz-coordinatesystem the unit vectors along the x-axis, y-axis and z-axis are denoted by i, j, and k,respectively.yz(0,1)(0,0,1)ji (1,0)xikj(0,1,0)y(1,0,0)Thus,xi = 〈1,0〉, j = 〈0,1〉 In 2-spacei = 〈1,0,0〉, j = 〈0,1,0〉, k = 〈0,0,1〉 In 3-space
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Chapter 1 Topics in Analytic Geometry

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