Exercises for Seetlon 13.2 Plot the points in Exercises 14. I. (1,0,0) 3. (3, 71,5) 2. (0,2,4) 4. (2, l,1l) Complete the computations in Exercises 58. 5. (6, 0, 5) + (5, 0,6) '- 6. (0,0,0) + (0, 0. 0) = 90:4 .'O 10. ll. 12. {l,3,5)+4{l, 3, -5)= (2:0: l)_ 8(3, D lbi' = Sketch v, 2v, and -v, where v has components (1, - I, - 1). Sketch v, 3v, and iv, where v has components (2, - l, 1). Let v have components (0, l, 1) and w have com- ponents (l, 1,0). Find v + w and sketch. Let v have components (2,-1.1) and w have components (1, l, 1). Find v + w and sketch. In Exercises [320, express the given vector in terms of the standard basis. 13. I4. 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 (e l, 2,11). The vector from (0, l, 2) to (l, l, l). The vector from (3,0, 5) to (2, 7.6). The vector from (1,0,0) to (2, l, l). The vector from (1,0,0) to (3, 2, 2). A ship at position (1,0) on a 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; I 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? (d) 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 + 500i 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 v1 is 40 miles per hour from east to west while an airplane travels with air speed ii: of 100 miles per hour due north. The speed of the airplane relative to the earth is the vector sum v. 4- v2. (a) Find v; + v2. (b) 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 1501bs, while B pulls with a force of HO 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 Exercises tor Section 13.3 Use vector methods in Exercises [6. 1. Show that the line segmentjoining the midpoints of two sides of a triangle is parallel to and has half of the length of the third side. . Prove that the medians of the triangle intersect in a point two-thirds of the way along any median from a vertex to the midpoint of the opposite side. . Prove that if PQR is a triangle in space and h > 0 is a number. then there is a triangle with sides parallel to those of PQR and side lengths 15 times those of PQR. Prove that if the corresponding sides of two triangles are parallel, then the lengths of corre- sponding sides have a common ratio. (Assume that the triangles are not degenerated into lines.) . Find the point in the plane two-thirds of the way from the origin to the midpoint of the line seg- ment between (1. I) and (2, -2). Let P = (3.5.2) and Q = (2.5.3). Find the point R such that Q is the midpoint of the line seg- ment PR. Write equations for the lines in Exercises 710. 3'. 8. 9. 10. The line through (1.1.0) and (0.0. l). The line through (2,0,0) and (0. 1.0). The line through (0, 0. 0) and (l, l, l). The line through (- I. 1,0) and (1.8. 4). Write parametric equations for the lines in Exercises 1 114. 12. 13. I4. 15. 16. 17. 18. The line through the point (1. 1,0) in the direc- tion of vector -i ~j + 1:. The line through (0. 1,0) in the direction j. The line in the plane through ( l. ~2) and in direction 3i - 2j. The line in the plane through (2. 1) and in direction i j. At what point does the line through (0, l, 2) with direction I + j + It cross the xy plane? Where does the line through (3.4.5) and (6.7.8) meet the yz plane? Do the lines given by R, = (i, 3! 1,41) and R2 = (3r.5. l - !) intersect? Find the unique value of c for which the lines R1 = (r, -6r + c.21~ 8) and R2 =(31 +1.2L0) intersect. on they axis? On 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,b.c), where a. b, and c are each equal to I or 1. Of what geometric figure are these the vertices? 29. Let u = 2i + 3j + It. Sketch the vectors in, 2n, and 3u on the same set of axes. In Exercises 3034, consider the vectors v = 3i + 4j + 5k and w = i j + k. Express the given vector in terms of i, j and It. 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 I) such that av + bw is the given vector. 35. i 36. j 37. 3i+ 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+ew=ri+sj+ tk. 39. A I-kilogram mass located at the origin is sus- pended by ropes attached to the points (1. I, l) and ( l, I, 1). 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? [Hirm Use the symmetry of the problem. A l-kilogram mass weighs 9.8 newtons.] 40. Write the chemical equation C0 + H20 = H2 + CD; as an equation in ordered triples, and illus- trate it by a vector diagram in space. 4|. (3) Write the chemical equation pC3H403 + qu = rC'Oz + SHJO 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 ri + sin tj + k at time t. (a) Draw the cardiac vector for r = 0, air/4, rr/Z, 311/4, 1r, 517/4, 311/2, ~Mir/4, 2%. (b) Describe the motion of the tip of the cardiac vector in space if the tail is fixed at the origin. +43. Let P, = (1,0,0) + r(2, I, I), where r 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 I". *44. The 2 coordinate of the point P in Fig. 13.2.I5 is 3. What are the x and y coordinates? Z Figure 13.2.15. Let P = (x, y, 3). What are x and y