SECTION 9.4 Vectors 647 Concepts and Vocabulary 9.4 Assess Your Understanding 1. A is a quantity that has both magnitude and direction. 2. If v is a vector, then ( ) + − = v v . 3. A vector u for which = u 1 is called a(n) vector. 4. If = 〈 〉 a b v , is an algebraic vector whose initial point is the origin, then v is called a(n) vector. 5. If = + a b v i j, then a is called the component of v and b is called the component of v . 6. If F1 and F2 are two forces acting on an object, the vector sum + F F 1 2 is called the force. 7. True or False Force is an example of a vector. 8. True or False Mass is an example of a vector. 9. Multiple Choice If v is a vector with initial point ( ) x y , 1 1 and terminal point ( ) x y , , 2 2 then which of the following is the position vector that equals v ? (a) 〈 − − 〉 x x y y , 2 1 2 1 (b) 〈 − − 〉 x x y y , 1 2 1 2 (c) − − x x y y 2 , 2 2 1 2 1 (d) + + x x y y 2 , 2 1 2 1 2 10. Multiple Choice If v is a nonzero vector with direction angle α α ° ≤ < ° , 0 360 , between v and i , then v equals which of the following? (a) α α ( ) − v i j cos sin (b) α α ( ) + v i j cos sin (c) α α ( ) − v i j sin cos (d) α α ( ) + v i j sin cos Historical Feature The history of vectors is surprisingly complicated for such a natural concept. In the xy -plane, complex numbers do a good job of imitating vectors. About 1840, mathematicians became interested in finding a system that would do for three dimensions what the complex numbers do for two dimensions. Hermann Grassmann (1809—1877), in Germany, and William Rowan Hamilton (1805—1865), in Ireland, both attempted to find solutions. Hamilton’s system was the quaternions , which are best thought of as a real number plus a vector; they do for four dimensions what complex numbers do for two dimensions. In this system the order of multiplication matters; that is, ≠ ab ba. Also, two products of vectors emerged, the scalar product (or dot product) and the vector product (or cross product). Grassmann’s abstract style, although easily read today, was almost impenetrable during the nineteenth century, and only a few of his ideas were appreciated. Among those few were the same scalar and vector products that Hamilton had found. About 1880, the American physicist Josiah Willard Gibbs (1839—1903) worked out an algebra involving only the simplest concepts: the vectors and the two products. He then added some calculus, and the resulting system was simple, flexible, and well adapted to expressing a large number of physical laws. This system remains in use essentially unchanged. Hamilton’s and Grassmann’s more extensive systems each gave birth to much interesting mathematics, but little of it is seen at elementary levels. Josiah Gibbs (1839–1903) Credit: Hulton Archive/ Stringer/Getty Images Skill Building In Problems 11–18, use the vectors in the figure at the right to graph each of the following vectors. 11. + v w 12. +u v 13. v3 14. w2 15. −v w 16. −u v 17. + − v u w 3 2 18. − + u v w 2 3 In Problems 19–26, use the figure at the right. Determine whether each statement given is true or false. 19. + = A B F 20. + = K G F 21. = − + C D E F 22. + + = G H E D 23. + = + E D G H 24. − = − H C G F 25. + + + = A B K G 0 26. + + + + = A B C H G 0 v u w G B F K A H C D E 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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