Review Exercise
1. Consider any line in space that does not pass through the origin.
a. Is it possible for this line to intersect just one coordinate axis? exactly
two? all three? none at all?
b. Is it possible for this line to intersect just one coordinate plane? exactly
two? all three? none at all?
2. Find a vector equation of the line
a. that passes through the points (3, 9) and (4, 2)
b. that passes through the point (5, 3) and is parallel to the line
៬r (4, 0) t(0, 5)
c. that is perpendicular to the line 2x 5y 6 0 and passes through the
point (0, 3)
3. Find parametric equations of the line
a. that passes through (9, 8) with slope 23
b. that passes through (3, 2) and is perpendicular to the line
r៬ (4, 1) t(3, 2)
c. through the points (4, 0) and (0, 2)
4. Find a vector equation of the line
a. that passes through the points (2, 0, 3) and (3, 2, 2)
b. that has an x-intercept of 7 and a z-intercept of 4
y2
x5
z6
and passes through the point
c. that is parallel to 4
5
2
(0, 6, 0)
5. Find parametric equations of the line
y2
x1
z 3 and passes through the
a. that is parallel to the line 3
2
origin
b. that passes through the point (6, 4, 5) and is parallel to the y-axis
c. that has a z-intercept of 3 and direction vector (1, 3, 6)
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6. Find the Cartesian equation of the line
a. that passes through the point (1, 2) and is parallel to the line
3x 4y 5 0
b. that passes through the point (7, 3) and is perpendicular to the line
x 2 t, y 3 2t
c. that passes through the origin and is perpendicular to the line
x 4y 1 0
7. a. Find the parametric equations of the line l that passes through the point
A(6, 4, 0) and is parallel to the line passing through B(2, 0, 4) and
C(3, 2, 1).
b. If (4, m, n) is a point on l, find m and n.
8. Determine if the following pairs of lines are parallel and distinct, coincident,
perpendicular, or none of these.
a. ៬r (2, 3) t(3, 1) and ៬r (1, 4) u(6, 2)
b. x 1 2t, y 3 t and x u, y 13 2u
y4
x1
, z 1 and x 4t, y 1 2t, z 6
c. 2
1
d. (x, y, z) (1, 7, 2) t(1, 1, 1) and (x, y, z) (3, 0, 1) u(2, 2, 2)
y6
x4
z2
meet the coordinate
9. At what points does the line 2
4
1
planes?
10. In the xy-plane,
a. find the Cartesian equation of the line r៬ (2, 3) t(1, 5)
b. find a vector equation of the line 5x 2y 10 0
c. find a vector equation of the line y 34x 12
11. Given the line ៬r (12, 8, 4) t(3, 4, 2),
a. find the intersections with the coordinate planes, if any
b. find the intercepts with the coordinate axes, if any
c. graph the line in an x-, y-, z-coordinate system
12. Find the direction cosines and the direction angles (to the nearest degree) of
the direction vectors of the following lines.
y6
x3
z1
a. 5
1
2
b. x 1 8t, y 2 t, z 4 4t
c. ៬r (7, 0, 0) t(4, 1, 0)
REVIEW EXERCISE
267
13. Find the intersection, if any, of
a. the line ៬r (0, 0, 2) t(4, 3, 4) and the line
៬r (4, 1, 0) u(4, 1, 2)
b. the line x t, y 1 2t, z 3 t and the line
x 3, y 6 2u, z 3 6u
14. Find the shortest distance between
a. the points (2, 1, 3) and (0, 4, 7)
b. the point (3, 7) and the line 2x 3y 7
c. the point (4, 0, 1) and the line ៬r (2, 2, 1) t(1, 2, 1)
y3
x1
z7
d. the point (1, 3, 2) and the line 1
2
1
15. Find the coordinates of the foot of the perpendicular from Q(3, 2, 4) to the
line r៬ (6, 7, 3) t(5, 3, 4).
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Chapter 7 Test
Achievement Category
Questions
Knowledge/Understanding
1, 4
Thinking/Inquiry/Problem Solving
5, 7
Communication
3
Application
2, 6
1. A line goes through the points (9, 2) and (3, 4). Determine
a. its vector equation
b. its parametric equations
c. its symmetric equation
d. its scalar equation
2. Find the scalar equation of the line which is perpendicular to the line
2x 3y 18 0 and has the same y-intercept as the line
(x, y) (0, 1) t(3, 4).
y4
x2
z2
with
3. Find any two of the three intersections of the line 6
3
3
the coordinate planes, and graph the line.
4. Find the distance from the point (1, 2, 3) to the line x y z 2.
5. A line through the origin has direction angles β 120º and γ 45º. Find a
vector equation for the line.
6. Determine the point of intersection of the two lines
y8
x5
z6
(x, y, z) (2, 0, 3) t(5, 1, 3) and 1
3
2
z
x1
y1
be two
7. Let l1:x 8 t, y 3 2t, z 8 3t and l2:
3
2
1
lines in three-dimensional space.
a. Show that l1 and l2 are skew lines (that is, neither parallel nor intersecting).
b. State the coordinates of P1, the point on l1 determined by t 2.
c. Determine the coordinates of P2, the point on l2 such that P1 P2 is perpendicular to l2.
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