Exercises for Sectlon 13.2 Plot the points in Exercises 14. 1. 3. (1,0,0) (3, l,5) 2. (0,2,4) 4- (2! _19%) Complete the computations in Exercises 58. 3 \"2'49\"! 10. ll. 12. mdn+mnm- mam+mam= (Lsn+qL1-n= (2.0.lJ-3t3.ri.i)= Sketch v, 2v, and -v, where v has components w-L-U Sketch v, 3v, and iv. where v has components (2, - l, 1). Let v have c0mponents (O, l, l) and w have com- ponents (1, 1.0). Find v + w and sketch. Let v have components (2. -1,1) and w have components (1, l, I). Find v + w and sketch. ln Exercises 1320, express the given vector in terms of the standard basis. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. The vector with components ( 1,2,3). The vector with components (0,2, 2). The vector with components (7,2,3). The vector with components ( l, 2.97). The vector from (0, 1,2) to (1, l, 1). The vector from (3,0, 5) to (2,7,6). The vector from (1,0,0) to (2, i l, l). The vector from (1,0. 0) to (3, 2, 2). A ship at position (1,0) on 21 nautical chart (with north in the positive y direction) sights a rock at position (2,4). What is the vector joining the ship to the rock? What angle does this vector make with due north? This is called the bearing of the rock from the ship. Suppose that the ship in Exercise 21 is pointing due north and travelling at a speed of 4 knots relative to the water. There is a current owing due east at 1 knot. (The units on the chart are nautical miles; 1 knot = 1 nautical mile per hour.) (a) If there were no current, what vector u would represent the velocity of the ship rela- tive to the sea bottom? 23. 24. 25. 26. (b) If the ship were just drifting with the cur rent, what vector v would represent its veloc- ity relative to the sea bottom? (c) What vector w represents the total velocity of the ship? (cl) Where would the ship be after 1 hour? (e) Should the captain change course? (f) What if the rock were an iceberg? An airplane is located at position (3, 4, 5) at noon and travelling with velocity 400i + 500] k kilo- meters per hour. The pilot spots an airport at position (23, 29,0). (a) At what time will the plane pass directly over the airport? (Assume that the earth is flat and that the vector It points straight up.) (b) How high abOve the airport will the plane be when it passes? The wind velocity v, is 40 miles per hour from east to west while an airplane travels with air speed v; of 100 miles per hour due north. The speed of the airplane relative to the earth is the vector sum v. + v2. (a) Find v, + v2. (13) Draw a figure to scale. A force of 50 lbs is directed 50 above horizon- tal, pointing to the right. Determine its horizontal and vertical components. Display all results in a figure. Two persons pull horizontally on ropes attached to a post, the angle between the ropes being 60. A pulls with a force of 150 lbs. while B pulls with a force of 110 lbs. (a) The resultant force is the vector sum of the two forces in a conveniently chosen coordi- nate system. Draw a figure to scale which graphically represents the three forces. (b) Using trigonometry, determine formulas for the vector components of the two forces in a conveniently chosen coordinate system. Per- form the algebraic addition, and find the angle the resultant force makes with A. 27. What restrictions must be placed on x, y, and z so that the triple (x,y,z) will represent a point 660 Chapter 13 Vectors on they axis? 0n the z axis? In the xy plane? In the x2 plane? 28. Plot on one set of axes the eight points of the form (a, 15.43), where a, b, and e are each equal to 1 or 1. Of what geometric gure are these the vertices? 29. Let u = 21 + 3j + k. Sketch the vectors u, 2u, and 3u on the same set of axes. In Exercises 3034, consider the vectors v = 3i + 4j + SI: and w = i i j + k. Express the given vector in terms of i. j and k. 30. v + w 31. 3v 32. 2w 33. 6v + SW 34. the vector u from the tip of w to the tip of v. (Assume that the tails of w and v are at the same point.) In Exercises 3537, let v = i +j and w = -i+j. Find numbers a and b such that av + bw is the given vector. 35. i 36. j 37. 31+ 7j 38. Let u=i+j+|t, v=i+j, and w=i. Given numbers r, s, and I, find a, b, and e such that au+bv+cw=ri+sj+ tit. 39. A l-kilogram mass located at the origin is sus- pended by ropes attached to the points (1,1,1) and ( l. 1,l). 1f the force of gravity is point- ing in the direction of the vector k, what is the vector describing the force along each rope? [Hahn Use the symmetry of the problem. A lwkilogram mass weighs 9.8 newtons.] 40. Write the chemical equation C0 + H20 =1'12 + CD; as an equation in ordered triples, and illus- trate it by a vector diagram in space. 41. (a) Write the chemical equation pC3H403 + qu = rCOz + sHZO as an equation in or- dered triples with unknown coefficients p, q. r. and s. (b) Find the smallest integer solution for p, q, r, and s. (c) Illustrate the solution by a vector diagram in space. 42. Suppose that the cardiac vector is given by cos xi + sin tj + It at time t. (a) Draw the cardiac vector for t = 0, 97/4, 51/2, Zia/4, tr, 517/4, 371/2, Mfr/4, 2n. (b) Describe the motion of the tip of the cardiac vector in space if the tail is fixed at the origin. 1:43. Let P, = (1,0,0) + t(2, l, I), where t is a real number. (a) Compute the coordinates of P, for r = l, 0, l, and 2. (b) Sketch these four points on the same set of axes (c) Try to describe geometrically the set of all the PF *44. The 2 coordinate of the point P in Fig. 13.2.15 is 3. What are the x and y coordinates? Z Figure 13.2.15. Let P = (x, y, 3). What are x and y? (1) Ex 13.2: #1319,#31,33;#35,37 (2) Ex 13.2: #39 (3) Ex 13.2: #21-25 (4) Consider the Cube in space with verticcs at (0, 0, 0), (1,0,0), (I, 1,0), (0,1,0), (0,0,1), (1,0, 1), (1,1,1), and (0, 1.1). Use vector methods to locate the point one-third of the way from the origin to the middle of the face whose vertices are(0,l.0), (0,1,1), (1,1,1), and (l. 1,0). (5) Explain velocity vectors in terms of displacement vectors and time. (7) Ex 13.3: #5-9 (8) Let v and w be two vectors in the xyz-space. (a) State the denition of v and w being parallel to each other. (b) Find two vectors v and W in the xyz-space (as ordered triples) and four distinct nonzero scalars a,b,c,d such that v and w have nonzero entries (as ordered triples) and av+ bw 2 cv + dw. If you gured out that this is not possible, prove that this is not possible. (9) Prove that the figure obtained by joining the midpoints of successive sides of any quadrilateral is a parallelogram. (10) Ex 13.3: #11,13; #15,17