678 Chapter 11: Vectors and the Geometry of Space
11.5 Lines and Planes in Space
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find a set of parametric equations of the line through the point parallel to
the vector .
a.
b.
c.
d.
e.
____ 2. Find a set of symmetric equations of the line through the point parallel to the
vector .
a.
b.
c.
d.
e.
____ 3. Find the set of parametric equations of the line through the point that is
parallel to the line , ,and .
a.
b.
c.
d.
e.
____ 4. Find a set of parametric equations of the line through the points and
.
a.
b.
c.
d.
e.
11.5 Lines and Planes in Space 679
____ 5. Find a set of symmetric equations of the line through the points and
.
a.
b.
c.
d.
e.
____ 6. Find a set of parametric equations of the line that passes through the point
and is parallel to the xy-plane and yz-plane.
a.
b.
c.
d.
e.
____ 7. Determine whether any of the lines given below are parallel or identical.
a. are identical
b. are identical
c. are identical
d. are parallel
e. are parallel
680 Chapter 11: Vectors and the Geometry of Space
____ 8. Determine whether any of the lines given below are parallel or identical.
a. are parallel
b. are identical
c. are parallel
d. are identical
e. are parallel
____ 9. Determine whether the lines given below intersect, and, if so, find the point of
intersection.
a. The lines intersect at .
b. The lines are parallel.
c. The lines intersect at .
d. The lines intersect at .
e. The lines are not parallel but do not intersect.
____ 10. Determine whether the lines given below intersect, and, if so, find the point of
intersection.
a. The lines are parallel.
b. The lines intersect at .
c. The lines intersect at .
d. The lines intersect at .
e. The lines are not parallel but do not intersect.
11.5 Lines and Planes in Space 681
____ 11. Find an equation of a plane passing through the given point and perpendicular to the
given vector.
Point: Vector:
a.
b.
c.
d.
e.
____ 12. Find an equation of a plane passing through the following three points.
a.
b.
c.
d.
e.
____ 13. Find an equation of a plane passing through the points
and perpendicular to the plane
.
a.
b.
c.
d.
e.
____ 14. Determine whether the following planes are parallel, orthogonal, or neither. If they
are neither parallel nor orthogonal, find the angle of intersection.
a. The planes are neither parallel nor orthogonal, the angle of intersection is .
b. The planes are parallel.
c. The planes are neither parallel nor orthogonal, the angle of intersection is .
d. The planes are orthogonal.
682 Chapter 11: Vectors and the Geometry of Space
____ 15. Determine whether the following planes are parallel, orthogonal, or neither. If they
are neither parallel nor orthogonal, find the angle of intersection.
a. The planes are neither parallel nor orthogonal, the angle of intersection is .
b. The planes are neither parallel nor orthogonal, the angle of intersection is .
c. The planes are orthogonal.
d. The planes are parallel.
____ 16. Find the distance between the point and the plane .
a.
b.
c.
d.
e.
____ 17. Find the distance between the planes given below.
a.
b.
c.
d.
e.
11.5 Lines and Planes in Space 683
____ 18. Find the distance between the point and the line given by the set of
parametric equations .
a.
522
/11
b. 2
c.
3110
/11
d. 4
e. 1
____ 19. A chute at the top of a grain elevator of a combine funnels the grain into a bin (see
figure). Find the angle between two adjacent sides. Round your answer to one decimal place.
a.
b.
c.
d.
e.
684 Chapter 11: Vectors and the Geometry of Space
____ 20. Two insects are crawling along different lines in three-space. At time t (in minutes),
the first insect is at the point where . Also, at time t, the second
insect is at the point where . Assume distances are given in inches.
Find the distance between the two insects at time . Round your answer to two decimal places.
a. inches
b. inches
c. inches
d. inches
e. inches
____ 21. Two insects are crawling along different lines in three-space. At time t (in minutes),
the first insect is at the point where . Also, at time t, the second
insect is at the point where . Assume distances are given in inches.
How close do the insects get? Round your answer to two decimal places.
a. inches
b. inches
c. inches
d. inches
e. inches
11.5 Lines and Planes in Space 685
11.5 Lines and Planes in Space – Answer Section