Review Exercise 1. Determine vector and parametric equations of the plane that contains the points A11, 2, 12, B12, 1, 12, and C13, 1, 42. 2. In question 1, there are a variety of different answers possible, depending on the points and direction vectors chosen. Determine two Cartesian equations for this plane using two different vector equations, and verify that these two equations are identical. 3. a. Determine the vector, parametric, and symmetric equations of the line passing through points A1 3, 2, 82 and B14, 3, 92. b. Determine the vector and parametric equations of the plane containing the points A1 3, 2, 82, B14, 3, 92, and C1 2, 1, 32. c. Explain why a symmetric equation cannot exist for a plane. 4. Determine the vector, parametric, and symmetric equations of the line passing through the point A17, 1, 22 and perpendicular to the plane with equation 2x 3y z 1 0. 5. Determine the Cartesian equation of each of the following planes: a. through the point P10, 1, 22, with normal n! 1 1, 3, 32 b. through the points 13, 0, 12 and 10, 1, 12, and perpendicular to the plane with equation x y z 1 0 c. through the points 11, 2, 12 and 12, 1, 42, and parallel to the x-axis 6. Determine the Cartesian equation of the plane that passes through the origin and contains the line r! 13, 7, 12 t12, 2, 32, t R. 7. Find the vector and parametric equations of the plane that is parallel to the yz-plane and contains the point A1 1, 2, 12. 8. Determine the Cartesian equation of the plane that contains the line r! 12, 3, 22 t11, 1, 42, t R, and the point 14, 3, 22. 9. Determine the Cartesian equation of the plane that contains the following lines: L 1 : r! and L 2 : r! 14, 4, 52 t15, 4, 62, t R, 14, 4, 52 s12, 3, 42, s R 10. Determine an equation for the line that is perpendicular to the plane 3x 2y z 1 passing through 12, 3, 32. Give your answer in vector, parametric, and symmetric form. 11. A plane has 3x 2y z 6 0 as its Cartesian equation. Determine the vector and parametric equations of this plane. 480 REVIEW EXERCISE
12. Determine an equation for the line that has the same x- and z-intercepts as the plane with equation 2x 5y z 7 0. Give your answer in vector, parametric, and symmetric form. 13. Determine the vector, parametric, and Cartesian forms of the equation of the plane containing the lines L and! 1 : r! 13, 4, 12 s11, 3, 52, s R, L 2 : r 2 17, 1, 02 t12, 6, 102, t R. 14. Sketch each of the following planes: a. p 1 : 2x 3y 6z 12 0 b. p 2 : 2x 3y 12 0 c. p 3 : x 3z 6 0 d. p 4 : y 2z 4 0 e. p 5 : 2x 3y 6z 0 15. Determine the vector, parametric, and Cartesian equations of each of the following planes: a. passing through the points and and parallel to the line with equation L : r! P11, 2ti! 2, 52 14t 32j! Q13, 1, 22 1t 12k!, t R b. containing the point A11, 1, 22 and perpendicular to the line joining the points B12, 1, 62 and C1 2, 1, 52 c. passing through the points 14, 1, 12 and 15, 2, 42 and parallel to the z-axis d. passing through the points 11, 3, 52, 12, 6, 42, and 13, 3, 32 16. Show that s, and L 2 : r! L 1 : r! 11, 2, 32 s1 3, 5, 212 t10, 1, 32, t R, 11, 1, 62 u11, 1, 12 v12, 5, 112, u, v R, are equations for the same plane. 17. The two lines and L 2 : r! L 1 : r! 1 1, 1, 02 s12, 1, 12, s R, 12, 1, 22 t12, 1, 12, t R, are parallel but do not coincide. The point A15, is on Determine the coordinates of a point B on such that AB! 4, 32 L 1. L 2 is perpendicular to L 2. 18. Write a brief description of each plane. a. p 1 : 2x 3y 6 b. p 2 : x 3z 6 c. p 3 : 2y z 6 19. a. Which of the following points lies on the line x 2t, y 3 t, z 1 t? A12, 4, 22 B1 2, 2, 12 C14, 5, 22 D16, 6, 22 b. If the point 1a, b, 32 lies on the line, determine the values of a and b. CHAPTER 8 481
20. Calculate the acute angle that is formed by the intersection of each pair of lines. a. L and L 2 : x 2 1 y 1 : x 1 y 3 1 5 2 3 b. y 4x 2 and y x 3 c. L 1 : x 1 3t, y 1 4t, z 2t and L 2 : x 1 2s, y 3s, z 7 s d. L 1 : 1x, y, z2 14, 7, 12 t14, 8, 42 and L 2 : 1x, y, z2 11, 5, 42 s1 1, 2,32 21. Calculate the acute angle that is formed by the intersection of each pair of planes. a. 2x 3y z 9 0 and x 2y 4 0 b. x y z 1 0 and 2x 3y z 4 0 22. a. Which of the following lines is parallel to the plane i. r! 4x y z 10 0? 13, 0, 22 t11, 2, 22 ii. x 3t, y 5 2t, z 10t x 1 iii. y 6 4 1 z 1 b. Do any of these lines lie in the plane in part a.? 23. Does the point (4, 5, 6) lie in the plane 1x, y, z2 14, 1, 62 p13, 2, 12 q1 6, 6, 12? Support your answer with the appropriate calculations. 24. Determine the parametric equations of the plane that contains the following two parallel lines: L 1 : 1x, y, z2 12, 4, 12 t13, 1, 12 and L 2 : 1x, y, z2 11, 4, 42 t1 6, 2, 22 25. Explain why the vector equation of a plane has two parameters, while the vector equation of a line has only one. 26. Explain why any plane with a vector equation of the form 1x, y, z2 1a, b, c2 s1d, e, f 2 t1a, b, c2 will always pass through the origin. 27. a. Explain why the three points 12, 3, 12, 18, 5, 52 and 1 1, 2, 12 do not determine a plane. b. Explain why the line r! 14, 9, 32 t11, 4, 22 and the point 18, 7, 52 do not determine a plane. 28. Find a formula for the scalar equation of a plane in terms of a, b, and c, where a, b, and c are the x-intercept, the y-intercept, and the z-intercept of a plane, respectively. Assume that all intercepts are nonzero. 482 REVIEW EXERCISE
29. Determine the Cartesian equation of the plane that has normal vector 16, 5, 122 and passes through the point 15, 8, 32. 30. A plane passes through the points A11, 3, 22, B1 2, 4, 22, and C13, 2, 12. a. Determine a vector equation of the plane. b. Determine a set of parametric equations of the plane. c. Determine the Cartesian equation of the plane. d. Determine if the point 13, 5, 42 lies on the plane. 31. Determine the Cartesian equation of the plane that is parallel to the plane 4x 2y 5z 10 0 and passes through each point below. a. 10, 0, 02 b. 1 1, 5, 12 c. 12, 2, 22 32. Show that the following pairs of lines intersect. Determine the coordinates of the point of intersection and the angles formed by the lines. a. L 1 : x 5 2t and L 2 : x 23 2s y 3 t y 6 s b. L and L 2 : x 6 y 2 1 : x 3 y 1 3 4 3 2 33. Determine the vector equation, parametric equations, and, if possible, symmetric equation of the line that passes through the point P11, 3, 52 and a. has direction vector 1 2, 4, 102 b. also passes through the point Q1 7, 9, 32 c. is parallel to the line that passes through R14, 8, 52 and S1 2, 5, 92 d. is parallel to the x-axis e. is perpendicular to the line 1x, y, z2 11, 0, 52 t1 3, 4, 62 f. is perpendicular to the plane determined by the points A14, 2, 12, B13, 4, 22, and C1 3, 2, 12 34. Determine the Cartesian equation of the plane that a. contains the point P1 2, 6, 12 and has normal vector 12, 4, 52 x 4 b. contains the point and the line y 2 z 1 P1 2, 0, 62 3 5 2 c. contains the point P13, 3, 32 and is parallel to the xy-plane d. contains the point P1 4, 2, 42 and is parallel to the plane 3x y 4z 8 0 e. is perpendicular to the yz-plane and has y-intercept 4 and z-intercept 2 f. is perpendicular to the plane x 2y z 6 and contains the line 1x, y, z2 12, 1, 12 t13, 1, 22 CHAPTER 8 483
Chapter 8 Test 1. a. Given the points A11, 2, 42, B12, 0, 32, and C14, 4, 42, i. determine the vector and parametric equations of the plane that contains these three points ii. determine the corresponding Cartesian equation of the plane that contains these three points b. Does the point with coordinates Q1, 1, 1 lie on this plane? 2 R 2. The plane p intersects the coordinate axes at 12, 0, 02, 10, 3, 02, and 10, 0, 42. x a. Write an equation for this plane, expressing it in the form a y b z 1. c b. Determine the coordinates of a normal to this plane. 3. a. Determine a vector equation for the plane containing the origin and the line with equation r! 12, 1, 32 t11, 2, 52, t R. b. Determine the corresponding Cartesian equation of this plane. 4. a. Determine a vector equation for the plane that contains the following two lines: L 1 : r! and L 2 : r! 14, 3, 52 t12, 0, 32, t R, 14, 3, 52 s15, 1, 12, s R b. Determine the corresponding Cartesian equation of this plane. x 2 5. a. A line has y 4 as its symmetric equations. Determine the 4 2 z coordinates of the point where this line intersects the yz-plane. b. Write a second symmetric equation for this line using the point you found in part a. 6. a. Determine the angle between p 1 and p 2 where the two planes are defined as p 1 : x y z 0 and p 2 : x y z 0. b. Given the planes p 3 : 2x y kz 5 and p 4 : kx 2y 8z 9, i. determine a value of k if these planes are parallel ii. determine a value of k if these planes are perpendicular c. Explain why the two given equations that contain the parameter k in part b cannot represent two identical planes. 7. a. Using a set of coordinate axes in R 2, sketch the line x 2y 0. b. Using a set of coordinate axes in R 3, sketch the plane x 2y 0. c. The equation Ax By 0, A, B 0, represents an equation of a plane in R 3. Explain why this plane must always contain the z-axis. 484 CHAPTER 8 TEST