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Exercises for Equations of Lines and Planes in Space
In exercises 1 - 4, points and are given. Let be the line passing through points and .
a. Find the vector equation of line .
b. Find parametric equations of line .
c. Find symmetric equations of line .
d. Find parametric equations of the line segment determined by and .
1)
2)
3)
4)
For exercises 5 - 8, point and vector are given. Let be the line passing through point with direction .
a. Find parametric equations of line .
b. Find symmetric equations of line .
c. Find the intersection of the line with the -plane.
5)
6)
7) where and
8) where and
For exercises 9 and 10, line is given.
a. Find a point that belongs to the line and a direction vector of the line. Express in component form.
b. Find the distance from the origin to line .
9)
10)
11) Find the distance between point and the line of symmetric equations
12) Find the distance between point and the line of parametric equations
For exercises 13 - 14, lines and are given.
a. Verify whether lines and are parallel.
b. If the lines and are parallel, then find the distance between them.
13)
14)
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15) Show that the line passing through points and is perpendicular to the line with equation
16) Are the lines of equations and perpendicular to
each other?
17) Find the point of intersection of the lines of equations and
18) Find the intersection point of the -axis with the line of parametric equations
For exercises 19 - 22, lines and are given. Determine whether the lines are equal, parallel but not equal, skew, or
intersecting.
19) and
20) and
21) and
22) and
23) Consider line of symmetric equations and point
a. Find parametric equations for a line parallel to that passes through point .
b. Find symmetric equations of a line skew to and that passes through point .
c. Find symmetric equations of a line that intersects and passes through point .
24) Consider line of parametric equations
a. Find parametric equations for a line parallel to that passes through the origin.
b. Find parametric equations of a line skew to that passes through the origin.
c. Find symmetric equations of a line that intersects and passes through the origin.
For exercises 25 - 28, point and vector are given.
a. Find the scalar equation of the plane that passes through and has normal vector .
b. Find the general form of the equation of the plane that passes through and has normal vector .
25)
26)
27)
28)
For exercises 29 - 32, the equation of a plane is given.
a. Find normal vector to the plane. Express using standard unit vectors.
b. Find the intersections of the plane with each of the coordinate axes (its intercepts).
c. Sketch the plane.
29) [T]
30)
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31)
32)
33) Given point and vector , find point on the -axis such that and are orthogonal.
34) Show there is no plane perpendicular to that passes through points and .
35) Find parametric equations of the line passing through point that is perpendicular to the plane of equation
36) Find symmetric equations of the line passing through point that is perpendicular to the plane of equation
37) Show that line is parallel to plane .
38) Find the real number such that the line of parametric equations is parallel to the
plane of equation
For exercises 39 - 42, the equations of two planes are given.
a. Determine whether the planes are parallel, orthogonal, or neither.
b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes.
Express the answer in degrees rounded to the nearest integer.
c. If the planes intersect, find the line of intersection of the planes, providing the parametric equations of this line.
39) [T]
40)
41)
42) [T]
For exercises 43 - 46, determine whether the given line intersects with the given plane. If they do intersect, state the
point of intersection.
43) Plane: Line:
44) Plane: Line:
45) Plane: Line:
46) Plane: Line:
47) Show that the lines of equations and are skew, and find the
distance between them.
48) Show that the lines of equations and
are skew, and find the distance between them.
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49)
Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in the air, curves ft to the right,
and falls ft away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular
to the ground, find its equation.
50) [T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates
(in meters) and (see the following figure).
a. Find the general form of the equation of the plane that contains the solar panel by using points and , and show
that its normal vector is equivalent to
b. Find parametric equations of line that passes through the center of the solar panel and has direction vector
which points toward the position of the Sun at a particular time of day.
c. Find symmetric equations of line that passes through the center of the solar panel and is perpendicular to it.
d. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines and .
Sheet No. 2