SECTION 9.5 The Dot Product 657 Solution Position the vectors in a coordinate system in such a way that the wagon is moved from 0, 0 ( ) to 100, 0 ( ) . The motion is from A 0, 0 ( ) = to B 100, 0 , ( ) = so AB i 100 . = The force vector F , as shown in Figure 77(b) on the previous page, is F i j i j i j 50 cos30 sin30 50 3 2 1 2 25 3 ( ) ( ) = ° + ° = + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + By formula (12), the work done is W AB F i j i 25 3 100 2500 3 foot-pounds ( ) = ⋅ = + ⋅ = Now Work PROBLEM 29 Historical Feature We stated in the Historical Feature in Section 9.4 that complex numbers were used as vectors in the plane before the general notion of a vector was clarified. Suppose that we make the correspondence a b a bi c d c di i j i j Vector Complex number ↔ + ↔ + + ↔ + Show that ( ) ( ) ( ) ( ) + ⋅ + = ⎡ ⎣⎢ + + ⎤⎦⎥ a b c d a bi c di i j i j real part This is how the dot product was found originally. The imaginary part is also interesting. It is a determinant (see Section 11.3) and represents the area of the parallelogram whose edges are the vectors. This is close to some of Hermann Grassmann’s ideas and is also connected with the scalar triple product of three-dimensional vectors. 1. In a triangle with sides a , b , c and angles A , B , C , the Law of Cosines states that . (p. 570) ‘Are You Prepared?’ The answer is given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 2. If a b v i j 1 1 = + and a b w i j 2 2 = + are two vectors, then the is defined as a a b b v w . 1 2 1 2 ⋅ = + 3. If v w 0, ⋅ = then the two vectors v and w are . 4. If v w3 , = then the two vectors v and w are . 5. True or False Given two nonzero, nonorthogonal vectors v and w , it is always possible to decompose v into two vectors, one parallel to w and the other orthogonal to w . 6. True or False Work is a physical example of a vector. 7. Multiple Choice The angle , 0 , θ θ π ≤ ≤ between two nonzero vectors u and v can be found using what formula? (a) u v sinθ = (b) u v cosθ = (c) u v u v sinθ = ⋅ (d) u v u v cosθ = ⋅ 8. Multiple Choice If two nonzero vectors v and w are orthogonal, then the angle between them has what measure? (a) π (b) 2 π (c) 3 2 π (d) 2π Concepts and Vocabulary 9. v i j w i j , = − = + 10. v i j w i j , = + = − + 11. v i j w i j 2 , 2 = + = − 12. v i j w i j 2 2 , 2 = + = + 13. v i j w i j 3 , = − = + 14. v i j w i j 3 , = + = − In Problems 9–18, (a) find the dot product v w; ⋅ (b) find the angle between v and w ; (c) state whether the vectors are parallel, orthogonal, or neither. Skill Building 15. v i j w i j 3 4 , 6 8 = + = − − 16. v i j w i j 3 4 , 9 12 = − = − 17. v i w j 4 , = = 18. v i w j , 3 = = − 19. Find a so that the vectors a v i j = − and w i j 2 3 = + are orthogonal. 20. Find b so that the vectors v i j = + and = +b w i j are orthogonal. 9.5 Assess Your Understanding 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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