15. Find the distance from (2, 8, -1) to the line through (1, 1, 1) in the direction of the vector (1/ 3 )i + (1/ V3 )j+ (1/3)k. 16. Find the distance from (1, 1, -1) to the line through (2, - 1, 2) in the direction of k. 17. Find the distance from (1, 1, 2) to the line x = 3t + 2, y = -t - 1, z = t+1. 18. Find the distance from (1, 1, 0) to the line through (1, 0, - 1) and (2, 3, 1).49. 50. 5]. Suppose that a force F (for example, gravity) is acting vertically downward on an object sitting on a plane which is inclined at an angle of 45 to the horizontal. Express this force as a sum of a force acting parallel to the plane and one acting perpendicular to it. Suppose that an object moving in direction i+ j is acted on by a force given by the vector 2i +j. Express this force as a sum of a force in the direction of motion and a force perpendicular to the direction of motion. A force of 6 newtons makes an angle of arr/4 radians with they axis, pointing to the right. The force acts against the movement of an object along the straight line connecting (l,2) to (5,4). (3) Find a formula for the force vector F. (b) Find the angle 6 between the displacement direction D = (5 1)i + (4 - 2)j and the force direction F. (c) The work done is F - D, or equivalently, ||F|l ||D||cos 0. Compute the work from both formulas and compare. 21. 22. 23. 24. 10. 11. 12. .(i+j+k)-(l+]+2k) .i-j 2. (i+j+k)-(i+k) 4. (31+4j)-(3j +4k) Find the angle between the pair of vectors in Exercise 1. Find the angle between the pair of vectors in Exercise 2. Find the angle between the pair of vectors in Exercise 3. Find the angle between the pair of vectors in Exercise 4. Find a unit vector in the xy plane which is orthogonal to 2i j. Find a unit vector in the xy plane which is orthogonal to 3] 5i. Use the formula (I+j+lr)'i=l to find the angle between the diagonal of a cube and one of its edges. Sketch. (a) Show that if ||u|l = 1M}, and u and v are not parallel, then u+v and uv are perpendic- ular. (b) Use the result of part (a) to prove that 14. 15. 16. 17. 18 Give mangeant mamerer, [S'a'rrgm rrrarrgre. . Show that the length of the orthogonal projection of v on u is equal to ||v|| |cos0|, where 9 is the angle between v and u. Use vector methods to prove that a triangle is isosceles if and only if its base angles are equal. Find the distance from (2,8, l) to the line through (1,1,1) in the direction of the vector (1/J3')i+(1/r)i+(I/r3')k. Find the distance from (l,],-l) to the line through (2, 1,2) in the direction of k. Find the distance from (1,1,2) to the line x =3r+2,y= :l.z=z+l. . Find the distance from (1,1,0) to the line through (1.0, l) and (2, 3, l). the equation for each of the planes in Exercises 1924. 19 20 . The plane through the origin orthogonal to the vector i + j + II. . The plane through (1,0,0) orthogonal to the vector i + j + It. Copyright 1985 Springer-Verlag. All rights reserved. The plane through the origin orthogonal to i. 13.4 The Dot Product 675 44. Find the following: The plane containing (11,17, c) with normal vector ai + bj + ck. The plane containing the points (0.0, 1), (1. 1, l) and (0, 1,0). The plane containing the points (1,0,0), (0,2,0), and (0, 0, 3). Find a unit vector orthogonal to each of the planes in Exercises 2528. 25. 26. 27. The plane given by 2x + 3y + z = 0. The plane given by 8x y 22 + 10 = 0. The plane through the origin containing the 45. 46. (a) A unit normal to the plane x 2y + z = 0. (b) A vector orthogonal to the vectors i j + k and i + j + k. (c) The angle between 2i + j + k and k i. (d) A vector in space making an angle of 45 with i and 60 with j. Let P. and P2 he points in the plane. Give an equation of the form ax + by = c for the perpen- dicular biscctor of the line segment between P. and P1. Given nonzero vectors a and I), show that the