Exercises for Section 13.1 Complete the computations in Exercises 1-4. 1. (1, 2) + (3, 7) = 2. (-2, 6) - 6(2, - 10) = 3. 3[(1, 1) - 2(3, 0)] = 4. 2[(8, 6) - 4(2, - 1)] = Solve for the unknown quantities, if possible, in Exer- cises 5-16. 5. (1, 2) + (0, y) = (1, 3) 6. r(7,3) = (14, 6) 7. a(2, - 1) = (6, -7) 8. (7,2) + (x, y) = (3, 10) 9. 2(1, b) + (b, 4) = (3, 4) 10. (x, 2) + (-3)(x, y) = (-2x, 1) 11. 0(3, a) = (3, a) 12. 6(1, 0) + b(0, 1) = (6,2) 13. a(1, 1) + b(1, - 1) = (3,5) 14. (a, 1) - (2, b) = (0,0) 15. (3a, b) + (b, a) = (1, 1) 16. a(3a, 1) + a(1, - 1) = (1,0)23. 24. 25. Al (I) What it me roex were an iceberg: An airplane is located at position (3,4, 5} at noon and travelling with velocity 400i + 500j 1: 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.) (In) 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 v1 + v2. (a) Find v1+ v2. (b) Draw a figure to scale. A force of 50 tbs is directed 50" above horizon- tal, pointing to the right. Determine its horizontal and vertical components. Display all results in a figure. '1"..._ ._____.__ .__.Il l___1_.__4..\"._ a... _.-.._....t noannLnA Exercises for Section 13.3 Use vector methods in Exercises 16. 1. Write 7. 8. 9. 10. Write 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 wag,r along any median from a vertex to the midpoint of the opposite side. . Prove that if PQR is a triangle in space and b > 0 is a number, then there is a triangle with sides parallel to these of PQR and side lengths :5 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. l) 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. equations for the lines in Exercises 710. The line through (1, [,0) and (0,0. l). The line through (2.0, 0) and (0. 1.0). The line through (0.0, 0) and (l, 1.1). The line through ( l. 1,0) and (1,8. 4). parametric equations for the lines in Exercises \"14. ll. 12. I3. 14. IS. 16. 1?. The line through the point {1. LO) in the direc- tion of vector i j + k. The line through (0, 1,0) in the direction j. The line in the plane through ( l, -2) and in direction 3i ~+ Zj. The line in the plane through (2, I) and in direction ij. At what point does the line through (0, 1,2) with direction i + j + k cross the xy plane? Where does the line through {3.4. 5} and {6,7, 8} meet the )2: plane? [)0 the lines given by RI = (L3! 1,4!) and R2 = (3r.5. l r} intersect? Co