EAGI uraca IUI OUDIIUI I I 0.0 Use vector methods in Exercises 16. Compute the length of the vectors in Exercises 19~24. 1, Show that the line segmentjoining the midpoints ;? | :i: k 32' ii +1. of two sides of a triangle is parallel to and has 23' iii 2k ' .H: J half of the length of the third side. ' + 24' l - J _ 3k 2. Prove that the medians of the triangle intersect in 25- For what a is llai _ 3i + k\" = 16? a point twothirds of the way along any median 26- For what 5 is \"i _ bi + 2k\" = 3? from a vertex to the midpoint of the opposite 27. Verify the triangle inequality for the vectors in side. Exercises 19 and 21. 3_ Prove that if pQR is a triangle in space and 28. Verify the triangle inequality for the vectors in b > 0 is a number, then there is a triangle with Exercises 20 and 22- sides parallel to those of PQR and side lengths b 29- Find unit \"\"0\" u, v, and W i" the plane SUCh times those of PQR. that u + 2" + W = 0- 4_ Prove that if the corresponding sides of two 30. Find unit vectors 1:, v, w, and z in the plane such triangles are parallel, then the lengths of corre- 1h\" u + " + W + 1 = 0- Sponding sides have a common ratio. (Assume 3'- Show that (0-0.1) (0.1.0); and (1,0,0) are the that the triangles are not degenerated into lines.) vertices Of an equilateral triangle. HOW 10118 is 5. Find the point in the plane two-thirds of the way eaCh Side? from the origin to the midpoint of the line seg 32- Find an equilateral triangle in space \"'hiCh ment between (1, 1) and (2. .4). shares just one side with the one in Exercise 31. 5_ Let p = (3, 5,2) and Q = (15,3). Find the point 33. Normalize the vectors in Exercises 19 and 2]. R such that Q is the midpoint of the line seg 34. Normalize the vectors in Exercises 20 and 22. merit pR. Findlthe distance between the pairs of points in Exer- cises 35-38. 35. (1,1,3) and (2,2,2) 3?. (1,1,2) and (1.2.3) 39. Let P, = r(3,2,1). (i) What is the distance from P, to (2,0, 0)? Write equations for the lines in Exercises 740. 7. The line through (1.1,0) and (0,0, 1). 8. The line through (2,0,0) and (0, 1, 0). 9. The line through (0,0,0) and (l, 1, l). 10. The line through ( 1, 1,0) and (1,8, -4). 36. (2.0.0) and (2, 1,2) 38. (1,2,3) and (3,2,1) Write parametric equations for the lines in Exercises 1 l 14. 11. The line through the point (I, 1.0) in the direc- tion of vector i j + IL. 40. (ii) For what value of r is the distance shortest? (iii) What is that shortest distance? Draw a figure, similar to Fig. 13.3.6, to illustrate the distance formula on p. 666. 12, The line through (0, 1, 0} in the direction j_ 41. A boat whose top speed in still water is 12 knots 13. The line in the plane through ( 1' ~2) and in points north and steams at full power. If there is direction 3i 2j, an eastward current of 5 knots, what is the speed 14. The line in the plane through (2. 1) and in of the boat? direction i j. 42. A ship starting at (0,0) proceeds at a speed of 10 15. At what point does the line through (0, 1,2) with knots directly toward a buoy located at (3'4)' direction i+l + k cross the xy plane? (The chart is measured in nautical miles; a knot 16. Where does the line through (3,4, 5) and (6, 7,8) equals I nautical mi]? per-hour.) meet the yz plane? (a) What IS the ships pomt of closest approach 17. Do the lines given by R. = (:.3: 1,40 and '0 a we" \"cam \"@273 _ _ R2 = (3! 5 l _ I) intersect? (b) After how long does it reach this potnt? l8. Find the unique value of c for which the lines (C) _How far {5 thispomt from the rock?- *43. Derive the perm-point form of the equation of a R. = (t, 6r + (321* 8) and R2 = (3! + 1,2r.0) intersect. line obtained in Section R.4 from the parametric