670 CHAPTER 9 Polar Coordinates; Vectors COMMENT Most graphing calculators can compute determinants that contain numerical entries. Figure 85 shows the calculation in Example 2 part (a) using a TI-84 Plus CE. Desmos also has a matrix calculator, not shown here. ■ Figure 85 Evaluating Determinants (a) 2 3 1 2 2 2 1 3 4 3 1 = ⋅ − ⋅ = − = (b) A B C A B C A B C A B C 2 3 5 1 2 3 3 5 2 3 2 5 1 3 2 3 1 2 9 10 6 5 4 3 ( ) ( ) ( ) = − + = − − − + − = − − + EXAMPLE 2 Now Work PROBLEM 7 The cross product of the vectors a b c v i j k 1 1 1 = + + and a b c w i j k, 2 2 2 = + + that is, b c b c a c a c a b a b v w i j k 1 2 2 1 1 2 2 1 1 2 2 1 ( ) ( ) ( ) × = − − − + − may be written symbolically using determinants as a b c a b c b c b c a c a c a b a b v w i j k i j k 1 1 1 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 × = = − + Using Determinants to Find Cross Products If v i j k 2 3 5 = + + and w i j k 2 3 , = + + find: (a) v w× (b) w v × (c) v v × (d) ×w w Solution EXAMPLE 3 (a) v w i j k i j k i j k 2 3 5 1 2 3 3 5 2 3 2 5 1 3 2 3 1 2 × = = − + = − − + (b) w v i j k i j k i j k 1 2 3 2 3 5 2 3 3 5 1 3 2 5 1 2 2 3 × = = − + = + − (c) v v i j k i j k i j k 2 3 5 2 3 5 3 5 3 5 2 5 2 5 2 3 2 3 0 0 0 0 × = = − + = − + = (d) w w i j k i j k i j k 1 2 3 1 2 3 2 3 2 3 1 3 1 3 1 2 1 2 0 0 0 0 × = = − + = − + = Now Work PROBLEM 15 2 Know Algebraic Properties of the Cross Product Notice in Examples 3(a) and (b) that ×v w and ×w v are negatives of one another. From Examples 3(c) and (d), one might conjecture that the cross product of a vector with itself is the zero vector.These and other algebraic properties of the cross product are given next.
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