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 a.
? c.
b. ____
d.
3. What is the point of intersection between
and ?
a.
c.
b.
d. 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? c. a. b.
____
6. Which of the following is a linear equation? a. b.
d.
c. d.
____
7. How many solutions are there to the system of equations a. 0 c. 3 b. 1 d. Infinity
____
8. What is the solution for the system of equations a. c. b.
____
d.
9. What is the solution to the following equations?
a.
c.
and
and
?
?
b.
d. none
____ 10. For what value of do the equations have infinitely many solutions?
a.
c.
b.
d. 100-2x
____ 11. What is the relationship between the planes a. intersect along a line b. coincident planes
and c. parallel planes d. other
?
____ 12. Which is not a solution to the following system?
a.
c.
b.
d.
____ 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 ? a. c. d. b.
and
____ 15. Which of the following planes produces a consistent system with the equations ? a. c. d. b.
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 and
R? a. b.
____ 18. For which value of k is the point
c. d.
a distance of four units from the plane
?
c. d.
a. b. ____ 19. Which point is equidistant from the planes a.
4
and
?
c.
b.
d.
____ 20. For which value of k are the planes
and
apart? a. 0 b. 1
a distance of
units
c. 2 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 distance between P and the point
intersects the plane
at point P. What is the
?
24. Solve the following system of equations.
25. Determine the solution to the system.
26. Determine the y-intercept for the line 27. Show that
so that it intersects the line
at
is a general solution for the linear equation
28. What is the direction vector of the line of intersection between the planes 29. What is the solution for the following system of equations?
. .
and
?
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
.
,
from point A to the line
, and
intersect at point A. Determine the distance .
38. What is the distance between the parallel planes 39. For what value of k is the point
and
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.
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 equations
and
47. Determine a point on the line is 8 units. 48. Determine the distance between the point , and
and the line of intersection between the two planes having . in which the minimal distance from the point to the line
and the plane determined by the points
,
.
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.
Unit 9 - Practice Test Answer Section MULTIPLE CHOICE 1. ANS: OBJ: 2. ANS: OBJ: 3. ANS: OBJ: 4. ANS: OBJ: 5. ANS: OBJ: 6. ANS: OBJ: 7. ANS: OBJ: 8. ANS: OBJ: 9. ANS: OBJ: 10. ANS: OBJ: 11. ANS: OBJ: 12. ANS: OBJ: 13. ANS: OBJ: 14. ANS: OBJ: 15. ANS: OBJ: 16. ANS: OBJ: 17. ANS: OBJ: 18. ANS: OBJ: 19. ANS: OBJ: 20. ANS: OBJ:
C PTS: 1 REF: Knowledge and Understanding 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines C PTS: 1 REF: Knowledge and Understanding 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines A PTS: 1 REF: Knowledge and Understanding 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines C PTS: 1 REF: Knowledge and Understanding 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines B PTS: 1 REF: Knowledge and Understanding 9.2 - Systems of Equations D PTS: 1 REF: Knowledge and Understanding 9.2 - Systems of Equations B PTS: 1 REF: Knowledge and Understanding 9.2 - Systems of Equations A PTS: 1 REF: Knowledge and Understanding 9.2 - Systems of Equations B PTS: 1 REF: Knowledge and Understanding 9.2 - Systems of Equations D PTS: 1 REF: Knowledge and Understanding 9.2 - Systems of Equations A PTS: 1 REF: Knowledge and Understanding 9.3 - The Intersection of Two Planes D PTS: 1 REF: Knowledge and Understanding 9.3 - The Intersection of Two Planes C PTS: 1 REF: Knowledge and Understanding 9.3 - The Intersection of Two Planes D PTS: 1 REF: Thinking 9.3 - The Intersection of Two Planes A PTS: 1 REF: Knowledge and Understanding 9.4 - The Intersection of Three Planes C PTS: 1 REF: Knowledge and Understanding 9.6 - The Distance from a Point to a Plane D PTS: 1 REF: Knowledge and Understanding 9.6 - The Distance from a Point to a Plane D PTS: 1 REF: Thinking 9.6 - The Distance from a Point to a Plane B PTS: 1 REF: Thinking 9.6 - The Distance from a Point to a Plane A PTS: 1 REF: Thinking 9.6 - The Distance from a Point to a Plane
SHORT ANSWER 21. ANS: None; the line is parallel to the plane. PTS: 1 REF: Knowledge and Understanding OBJ: 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines 22. ANS:
PTS: 1 REF: Application OBJ: 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines 23. ANS: 9 PTS: 1 REF: Application OBJ: 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines 24. ANS: No solutions; the system is inconsistent. PTS: 1 25. ANS:
REF: Knowledge and Understanding
PTS: 1 REF: Thinking 26. ANS: , this gives the point of intersection
OBJ: 9.2 - Systems of Equations
OBJ: 9.2 - Systems of Equations
PTS: 1 REF: Application OBJ: 9.2 - Systems of Equations 27. ANS: By substituting x and y into the equation and simplifying, we find that the solution works. PTS: 1 28. ANS:
REF: Application
OBJ: 9.2 - Systems of Equations
PTS: 1 REF: Knowledge and Understanding OBJ: 9.3 - The Intersection of Two Planes 29. ANS: Answers may vary. For example: PTS: 1 REF: Knowledge and Understanding OBJ: 9.3 - The Intersection of Two Planes 30. ANS:
PTS: 1 REF: Knowledge and Understanding OBJ: 9.3 - The Intersection of Two Planes 31. ANS: The planes intersect at the point . PTS: 1 REF: Knowledge and Understanding OBJ: 9.4 - The Intersection of Three Planes 32. ANS: The planes intersect at a single point, , the origin. PTS: 1 33. ANS:
REF: Application
OBJ: 9.4 - The Intersection of Three Planes
PTS: 1 REF: Knowledge and Understanding OBJ: 9.5 - The Distance from a Point to a Line in R^2 and R^3 34. ANS:
PTS: 1 35. ANS:
REF: Application
OBJ: 9.5 - The Distance from a Point to a Line in R^2 and R^3
PTS: 1 REF: Knowledge and Understanding OBJ: 9.5 - The Distance from a Point to a Line in R^2 and R^3 36. ANS:
PTS: 1 37. ANS:
REF: Application
OBJ: 9.5 - The Distance from a Point to a Line in R^2 and R^3
PTS: 1 38. ANS:
REF: Application
OBJ: 9.5 - The Distance from a Point to a Line in R^2 and R^3
PTS: 1 REF: Knowledge and Understanding OBJ: 9.6 - The Distance from a Point to a Plane 39. ANS: or PTS: 1
REF: Thinking
OBJ: 9.6 - The Distance from a Point to a Plane
40. ANS: or PTS: 1
REF: Thinking
OBJ: 9.6 - The Distance from a Point to a Plane
PROBLEM 41. ANS: Substituting
and
into the equation for the plane, we find that both of the lines lie on
the plane. PTS: 1 REF: Communication OBJ: 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines 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 . PTS: 1 REF: Communication OBJ: 9.2 - Systems of Equations 43. ANS: The two linear equations have different slopes, so there will be one solution regardless of what and are. PTS: 1 REF: Communication OBJ: 9.2 - Systems of Equations 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
.
PTS: 1 REF: Application OBJ: 9.2 - Systems of Equations 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. PTS: 1 REF: Thinking OBJ: 9.4 - The Intersection of Three Planes 46. ANS: . Using the distance The line of intersection between the two planes is formula, we find that the distance from the point to this line is PTS: 1 47. ANS:
REF: Application
.
OBJ: 9.5 - The Distance from a Point to a Line in R^2 and R^3
, therefore
must equal either 46 or
equation out of it, we can solve it with our given equation
. Taking either value and making an to find the required point,
or PTS: 1 48. ANS:
REF: Thinking
By computing the cross product of
OBJ: 9.5 - The Distance from a Point to a Line in R^2 and R^3 and
, we determine the plane to have equation
. Using the distance formula, the required distance is PTS: 1 49. ANS:
REF: Application
.
OBJ: 9.6 - The Distance from a Point to a Plane
PTS: 1 REF: Application OBJ: 9.6 - The Distance from a Point to a Plane 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. PTS: 1 REF: Communication OBJ: 9.6 - The Distance from a Point to a Plane