662 CHAPTER 9 Polar Coordinates; Vectors Finding a Position Vector Find the position vector of the vector PP v 1 2 = if P 1, 2, 3 1 ( ) = − and P 4, 6, 2 . 2 ( ) = Solution EXAMPLE 2 By equation (2), the position vector equal to v is v i j k i j k 4 1 6 2 2 3 5 4 ( ) [ ] ( ) ( ) =−− +− +− =+− Adding and Subtracting Vectors If v i j k 2 3 2 = + − and w i j k 3 4 5 , = − + find: (a) v w + (b) v w− Solution EXAMPLE 3 (a) v w i j k i j k i j k i j k 2 3 2 3 4 5 2 3 3 4 2 5 5 3 ( ) ( ) ( ) ( ) ( ) + = + − + − + = + + − + − + = − + (b) v w i j k i j k i j k i j k 2 3 2 3 4 5 2 3 3 4 2 5 7 7 ( ) ( ) ( ) ( ) [ ] [ ] − = + − − − + = − + − − + − − = − + − Finding Scalar Products and Magnitudes If v i j k 2 3 2 = + − and w i j k 3 4 5 , = − + find: (a) v3 (b) v w 2 3 − (c) v Solution EXAMPLE 4 (a) v i j k i j k 3 3 2 3 2 6 9 6 ( ) = + − = + − (b) v w i j k i j k 2 3 2232 3345 ( ) ( ) − = + − − − + i j k i j k i j k 4 6 4 9 12 15 5 18 19 =+−−+ − =−+ − (c) v i j k 2 3 2 2 3 2 17 2 2 2 ( ) = + − = + + − = Now Work PROBLEM 29 DEFINITION Let a b c v i j k 1 1 1 = + + and a b c w i j k 2 2 2 = + + be two vectors, and let α be a scalar. Then • a a b b c c v w if and only if , , and 1 2 1 2 1 2 = = = = • a a b b c c v w i j k 1 2 1 2 1 2 ( ) ( ) ( ) + = + + + + + • a a b b c c v w i j k 1 2 1 2 1 2 ( ) ( ) ( ) − = − + − + − • a b c v i j k 1 1 1 α α α α ( ) ( ) ( ) = + + • a b c v 1 2 1 2 1 2 = + + 3 Perform Operations on Vectors Equality, addition, subtraction, scalar product, and magnitude can be defined in terms of the components of a vector. These definitions are compatible with the geometric definitions given in Section 9.4 for vectors in a plane. Now Work PROBLEMS 33 AND 39
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