11 how do you nd a vector perpendicular to a plane

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Unformatted text preview: mining whether three points P, Q, and R lie on the same line. (b) Describe a method for determining whether four points P, Q, R, and S lie in the same plane. 17. (a) How do you find the distance from a point to a line? (b) How do you find the distance from a point to a plane? (c) How do you find the distance between two lines? 18. What are the traces of a surface? How do you find them? 19. Write equations in standard form of the six types of quadric surfaces. 20. (a) Write the equations for converting from cylindrical to rectangular coordinates. In what situation would you use cylindrical coordinates? (b) Write the equations for converting from spherical to rectangular coordinates. In what situation would you use spherical coordinates? 5E-13(pp 878-883) 1/18/06 11:46 AM Page 881 CHAPTER 13 REVIEW ■ TRUE-FALSE QUIZ 2. For any vectors u and v in V3, u 3. For any vectors u and v in V3, u u u. v ku v v w w v 4x 6y EXERCISES 10 z u 0. v u v. D 0 represents a line 1} is a circle. y2 v1, v2 , then u u1v1, u2 v2 . v ■ 2 x, 4, x are orthogonal. 6. Find two unit vectors that are orthogonal to both j 2 and i 0 (b) a b (d) 2 a b 2k 3 k. 2j 7. Suppose that u v w (a) u v w (c) v u w 2. Find (b) u w (d) u v v v 8. Show that if a, b, and c are in V3, then b b c c a a b c 2 9. Find the acute angle between two diagonals of a cube. a b 10. Given the points A 1, 0, 1 , B 2, 3, 0 , C 1, 1, 4 , and D 0, 3, 2 , find the volume of the parallelepiped with adjacent edges AB, AC, and AD. 3. If u and v are the vectors shown in the figure, find u v and v . Is u Cz By u1, u2 and v a u w. 5. Find the values of x such that the vectors 3, 2, x and 1, 2 and 2. Copy the vectors in the figure and use them to draw each of the following vectors. (a) a b (c) 1 a 2 v v 14. If u w. radius 3. (b) Find the center and radius of the sphere z2 u 13. The set of points { x, y, z x 2 1. (a) Find an equation of the sphere with center 1, y2 w in space. ■ x2 v v v. u w. 12. A linear equation Ax 6. For any vectors u, v, and w in V3, u v 11. The cross product of two unit vectors is a unit vector. v. ku u 10. For any vectors u and v in V...
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