640 CHAPTER 9 Polar Coordinates; Vectors 2 Find a Position Vector Up to this point, we have viewed vectors graphically. In applications it is advantageous to represent a vector algebraically. A rectangular coordinate system is used to represent algebraic vectors in the plane. If = 〈 〉 a b v , is an algebraic vector whose initial point is at the origin, then v is called a position vector . See Figure 56. Notice that the terminal point of the position vector = 〈 〉 a b v , is the point ( ) = P a b , . Figure 56 Position vector v y P 5 (a, b) x o v = 8a, b9 The next theorem states that any vector whose initial point is not at the origin is equal to a unique position vector. THEOREM Suppose that v is a vector with initial point ( ) = P x y , , 1 1 1 not necessarily the origin, and terminal point ( ) = P x y , . 2 2 2 If = P P v , 1 2 then v is equal to the position vector = 〈 − − 〉 x x y y v , 2 1 2 1 (1) To see why this is true, look at Figure 57. Figure 57 = 〈 〉 = 〈 − − 〉 a b x x y y v , , 2 1 2 1 a b v P 5 (a, b) a A v P2 5 (x2, y2) P1 5 (x1, y1) x2 2 x1 y2 2 y1 Q O b x y Triangle OPA and triangle PPQ 1 2 are congruent. [Do you see why? The line segments have the same magnitude, so ( ) ( ) = d O P d P P , , ; 1 2 and they have the same direction, so ∠ = ∠ POA P PQ. 2 1 Since the triangles are right triangles, we have angle–side–angle.] It follows that corresponding sides are equal. As a result, − = x x a 2 1 and − = y y b, 2 1 so v may be written as = 〈 〉 = 〈 − − 〉 a b x x y y v , , 2 1 2 1 Because of this result, any algebraic vector can be replaced by a unique position vector, and vice versa.This flexibility is one of the main reasons for the wide use of vectors. DEFINITION Algebraic Vector An algebraic vector v is represented as = 〈 〉 a b v , where a and b are real numbers (scalars) called the components of the vector v .
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