Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 38Every vector in 2-space is expressible uniquely in terms of i and j, and every vector in3-space is expressible uniquely in terms of i, j, and k as follows:Example 3.8v = 〈v 1 ,v 2 〉 = 〈v 1 ,0〉+〈0,v 2 〉 = v 1 〈1,0〉+v 2 〈0,1〉 = v 1 i+v 2 jv = 〈v 1 ,v 2 ,v 3 〉 = v 1 〈1,0,0〉+v 2 〈0,1,0〉+v 3 〈0,0,1〉 = v 1 i+v 2 j+v 3 k2-space〈3,−4〉 = 3i−4j3-space〈2,3,−5〉 = 2i+3j−5k〈5,0〉 = 5i+0j = 5i 〈0,0,3〉 = 3k〈0,0〉 = 0i+0j = 0 〈0,0,0〉 = 0i+0j+0k = 0(3i−2j)+(i+4j) = 2i+2j3(2i−4j) = 6i−12j(2i−j+3k)+(3i+2j−k) = i+j+2k2(3i+4j−k) = 6i+8j−2k‖3i+4j‖ = √ 3 2 +4 2 = 5 ‖2i−j+3k‖ = √ 2 2 +(−1) 2 +3 2 = √ 14‖v 1 i+v 2 j‖ = √ v 2 1 +v 2 2 ‖〈v 1 ,v 2 ,v 3 〉‖ = √ v 2 1 +v 2 2 +v 2 3Normalizing a VectorA common problem in applications is to find a unit vector u that has the same directionas some given nonzero vector v. This can be done by multiplying v by the reciprocal of itslength; that is,u = 1‖v‖ v = v‖v‖is a unit vector with the same direction as v.The process of multiplying v by the reciprocal of its length to obtain a unit vector withthe same direction is called normalizing v.Example 3.9 Find the unit vector that has the same direction asSolution .........v = 2i−j−2k.Vectors Determined by Length and AngleIf v is a nonzero vector with its initial point at the origin of an xy-coordinate system, andif θ is the angle from the positive x-axis to the radial line through v, then the x-componentof v can be written as ‖v‖cosθ and the y-component as ‖v‖sinθ;