652 Chapter 13 Vectors (b) Write the equation in part (a) as a pair of simultaneous equations in k, I. and m. (c) Solve the equation in part (b) for the small est positive integer values of k, I, and m. 26. Illustrate the solution of Exercise 25 by a vector diagram in the plane, with 503, 52. and SO; represented as vectors. , 27. In Fig. 13.1.15, which vector is (a) a 13?. (b) i\"? 28. In Fig. 13.1.15, find the number r such that c a = 1'1). 29. Trace Fig. 13.1.15 and draw the vectors (a) c + d, (b) 2e + a. What are their components? 30. Trace Fig. 13.1.15 and draw the vectors (a) 3(e - d), (b) c. What are their components? .\\/ \\ -34$67891011x uw-uuc~'- b Figure 13.1.15. Compute with these vectors in Exercises 2730. In Exercises 31736, let u have components (2, l) and v have components (1,2). Draw each of the indicated vectors. 31. u+ v 32. uiv 33. 2u 34. 4v 35. 2u-4v 36. u+2v 37. Let P = (2, I), Q = (3, 3), and R = (4, 1) be points in the xy plane. (a) Draw (on the same diagram) these vectors: v joining P to Q; wjoining Q to R; ujoining R to P. (b) What are the components of v, w, and u? (c) Whatisv+w+u? 38. Answer the questions in Exercise 37 for P = (-2, I), Q =(3, -3), and R =( 1, 4). 39. *41. *42. (a) Draw the vector v1 joining (1,0) to (1, l). (b) What are the components of n? (c) Draw v2 joining (1.0) to (13) and find the components of VI. (d) Draw the vector vJ joining (1,0) to (I, 2). (e) What are the coordinates of an arbitrary point on the vertical (that is. parallel to they axis) line through (1,0)? (f) What are the components of the vector v joining (1,0) to such a point? (3.) Draw a vector v joining ( 1. I) to (l, l). (b) What are the components of v? (c) Sketch the vectors v, :(1,l)+ (v when (=0, %, g, g, and l. (d) Describe, geometrically, the set of vectors v, = ( l, 1) + rv, where t takes on all values between 0 and 1. (Assume that all the vec- tors have their tails at the origin.) We say that v and w are linearly dependent if there are numbers r and s, not both zero, such that rv + .rw =0. Otherwise v and w are called linearly independent. (a) Are (0,0) and (1, 1) linearly dependent? (b) Show that two non-zero vectors are linearly dependent if and only if they are parallel. (e) Let v and w be vectors in the plane given by v = (a,b) and w = (c.d). Show that v and w are linearly dependent if and only if ad = bc. [Hint For one implication, you might use three cases: it #0, (13* 0. and b = d = 0.] ((1) Suppose that v and w are vectors in the plane which are linearly independent. Show that for any vector u in the plane there are numbers x and y such that xv + yw = u. Let P=(a,b) and Q = (c.d) be points in the plane. (You may assume that 0 :1 >0 to make the picture unambiguous.) Compute the area of the parallelogram with ver- tices at 0, P. Q. and P+ Q. Comment on the relationship between this and Exercise 41(d). Exercises for Section 13.2 Plot the points in Exercises l-4. l. 3. (1,0,0) 2. (0,2,4) (3, I,5) 4. (2, 1,.g,) Complete the computations in Exercises 58. 5. . (0,0,0) + (0, 0, 0) = .'9 91:49 10. ll. 12. (6.0, 5) + (5. 0, 6) = (19395)+ 4(_ la _39 -S) = (2,0, 1) - 3(3, 9 = Sketch v, 2v, and v, where v has components (1, l, 1). Sketch v, 3v, and )v, where v has components (2, l, 1). Let v have components (0, 1, l) and w have com- ponents (1, 1,0). Find v + w and sketch. Let v have components (2, -1, I) and w have components (I, l, I). Find v + w and sketch