Unit 9 - Practice Test

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1 Unit 9 - Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of the following is not a possible number of intersections between a line and a plane? a. 0 c. 2 b. 1 d. 2. What is the normal vector of the plane? 3. What is the point of intersection between and? none 4. If two lines have no points of intersection and the same direction vector, they are: a. intersecting lines c. parallel lines b. skew lines d. coincident lines 5. Which of the following is not a linear equation? 6. Which of the following is a linear equation? 7. How many solutions are there to the system of equations and? a. 0 c. 3 b. 1 d. Infinity 8. What is the solution for the system of equations and? 9. What is the solution to the following equations?

2 none 10. For what value of do the equations have infinitely many solutions? 100-2x 11. What is the relationship between the planes and? a. intersect along a line c. parallel planes b. coincident planes d. other 12. Which is not a solution to the following system? 13. What is the nature of intersection between the planes and? a. parallel planes c. intersection at a line b. coincident planes d. other 14. Which of the following is a line parallel to the line intersecting the planes? 15. Which of the following planes produces a consistent system with the equations? and and 16. The distance between two skew lines: a. is the same for all points on each line b. is the same for each point on one line and a unique point on the other c. is the shortest at a unique point on each line d. none of the above 17. What is the distance between the skew lines R? R and 18. For which value of k is the point a distance of four units from the plane?

3 4 19. Which point is equidistant from the planes and? 20. For which value of k are the planes and a distance of units apart? a. 0 c. 2 b. 1 d. 3 Short Answer 21. Determine the point of intersection between the line and the plane. 22. The line crosses the xz-plane and the yz-plane at points A and B. What is the length of the segment connecting A and B? 23. The line intersects the plane at point P. What is the distance between P and the point? 24. Solve the following system of equations. 25. Determine the solution to the system. 26. Determine the y-intercept for the line so that it intersects the line at. 27. Show that is a general solution for the linear equation. 28. What is the direction vector of the line of intersection between the planes and? 29. What is the solution for the following system of equations?

4 30. Show that the line lies on the plane. 31. Solve the following system of equations. 32. Determine the intersection of the xy-plane, the yz-plane, and the xz-plane. 33. Determine the distance between the parallel lines and. 34. Calculate the distance between the x-axis and the point. 35. Determine the distance between the point and the line. 36. Determine the coordinate on the line which is the shortest distance from the line to the point. 37. The planes,, and intersect at point A. Determine the distance from point A to the line. 38. What is the distance between the parallel planes and? 39. For what value of k is the point a distance of 8 units form the plane? 40. For what values of k are the planes and a distance of apart? Problem 41. Show that the lines and lie on the plane. 42. Determine values for for which the following system has one solution, no solutions, and an infinite number of solutions.

5 43. a, b R How many solutions will the system have and why? 44. Two lines with slopes and intersect at. Determine the equations of the two lines and check your answer by solving them. 45. Considering consistent systems only, which type of intersection is possible with three planes and not possible with two planes? Explain. 46. Compute the distance between the point and the line of intersection between the two planes having equations and. 47. Determine a point on the line in which the minimal distance from the point to the line is 8 units. 48. Determine the distance between the point and the plane determined by the points,, and. 49. Two lines have equations and. What is the minimal distance between the two lines? 50. Explain why we use the distance formula from a point to a plane to figure the distance between two parallel planes.

6 Unit 9 - Practice Test Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: C 3. ANS: A 4. ANS: C 5. ANS: B 6. ANS: D 7. ANS: B 8. ANS: A 9. ANS: B 10. ANS: D 11. ANS: A 12. ANS: D 13. ANS: C 14. ANS: D REF: Thinking 15. ANS: A OBJ: The Intersection of Three Planes 16. ANS: C 17. ANS: D 18. ANS: D REF: Thinking 19. ANS: B REF: Thinking 20. ANS: A REF: Thinking

7 SHORT ANSWER 21. ANS: None; the line is parallel to the plane. 22. ANS: REF: Application 23. ANS: 9 REF: Application 24. ANS: No solutions; the system is inconsistent. 25. ANS: REF: Thinking 26. ANS:, this gives the point of intersection REF: Application 27. ANS: By substituting x and y into the equation and simplifying, we find that the solution works. REF: Application 28. ANS: 29. ANS: Answers may vary. For example: 30. ANS:

8 31. ANS: The planes intersect at the point. OBJ: The Intersection of Three Planes 32. ANS: The planes intersect at a single point,, the origin. REF: Application OBJ: The Intersection of Three Planes 33. ANS: OBJ: The Distance from a Point to a Line in R^2 and R^3 34. ANS: REF: Application OBJ: The Distance from a Point to a Line in R^2 and R^3 35. ANS: OBJ: The Distance from a Point to a Line in R^2 and R^3 36. ANS: REF: Application OBJ: The Distance from a Point to a Line in R^2 and R^3 37. ANS: REF: Application OBJ: The Distance from a Point to a Line in R^2 and R^3 38. ANS: 39. ANS: or REF: Thinking

9 40. ANS: or REF: Thinking PROBLEM 41. ANS: Substituting and into the equation for the plane, we find that both of the lines lie on the plane. REF: Communication 42. ANS: Case 1: The system has a unique solution when R and. Case 2: The system has no solutions when R,. Case 3: The system has an infinite amount of solutions when. REF: Communication 43. ANS: The two linear equations have different slopes, so there will be one solution regardless of what and are. REF: Communication 44. ANS: Given the slope and a point on the lines, we determine the lines to be and. After solving these equations, we find they intersect at the point. REF: Application 45. ANS: Intersection at a point is the type of intersection that is not possible with two planes. Planes extend indefinitely in all directions, so there is no way to have two planes intersect at a point. They either intersect in a line or are parallel. Three planes can easily intersect at a single point. REF: Thinking OBJ: The Intersection of Three Planes 46. ANS: The line of intersection between the two planes is. Using the distance formula, we find that the distance from the point to this line is. REF: Application OBJ: The Distance from a Point to a Line in R^2 and R^3 47. ANS:

10 , therefore must equal either 46 or. Taking either value and making an equation out of it, we can solve it with our given equation or to find the required point, REF: Thinking OBJ: The Distance from a Point to a Line in R^2 and R^3 48. ANS: By computing the cross product of and, we determine the plane to have equation. Using the distance formula, the required distance is. REF: Application 49. ANS: REF: Application 50. ANS: Since the two planes are parallel, they are the same distance apart at every point. This distance is the minimal distance. Because of this, we can choose any point on one plane and use the distance formula to obtain the minimum distance from the other plane. REF: Communication

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