Exercise 7.1
Part A
Communication
Knowledge/
Understanding
1. What is a direction vector? What is a parameter? What role do these quantities play in the equation of a line?
2. State a direction vector for each of the following lines.
a. a line parallel to x 9 3t, y 4 t
b. the line through (6, 4) and (2, 6)
c. the line y 3x 6
d. a line parallel to ៬r (1, 7) t(4, 3)
e. a horizontal line
f. a vertical line
3. State the coordinates of two points on each of the following lines.
a. x 3 8t, y 4t
b. ៬r (4, 0) t(0, 5)
Knowledge/
Understanding
4. State a parametric equation and a vector equation for each of the following
lines.
a.
b.
y
y
x
x
5. Graph the following lines.
b. ៬r (3, 4) t(4, 3)
a. x 1 5t
y 6 2t
6. For each of the following, find the parametric equations of the line that passes
through the point P with direction vector ៮៬
d. In each case, find two points on
the line different from P.
b. P(5, 0), ៮៬
d (1, 3)
a. P(1, 1), d៮៬ (4, 4)
7. State parametric equations
a. for the x-axis
b. for a line parallel to but not coincident with the x-axis
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Part B
8. For each of the following lines, find the vector equation that passes through
the point P with direction vector d៮៬.
b. P冢2, 3冣, d៮៬ 冢2, 6冣
a. P(2, 7), ៮៬
d (3, 4)
4
苶, 3)
c. P(1, 1), ៮៬
d (兹3
Thinking/Inquiry/
Problem Solving
3
d. P(0, 0), d៮៬ (2, 3)
9. For each of the following lines, state a direction vector with integer components. If possible, name a point on the line with integer coordinates.
a. x 13 2t, y 3 23t
b. ៬r 冢13, 12冣 t 冢13, 14冣
c. ៬r 冢12, 3冣 t 冢12, 5冣
Thinking/Inquiry/
Problem Solving
10. For each of the following, determine which pairs of lines are parallel and
which are perpendicular.
a. x 1 3t, y 7 4t and x 2 4s, y 3s
b. ៬r (1, 7) t(3, 4) and ៬r (2, 0) s(3, 4)
c. ៬r (1, 7) t(3, 4) and ៬r (2, 0) s(4, 3)
11. Find a vector equation of the line that passes through the point (4, 5) and is
perpendicular to the line ៬r (1, 8) t(3, 7).
12. Find the points where each of the following lines intersects the x- and y-axes.
Graph the line.
a. x 6, y 1 7t
b. ៬r (5, 10) t(1, 5)
c. ៬r (2, 3) t(3, 1)
Application
13. Show that both lines ៬r (3, 9) t (2, 5) and ៬r (5, 6) u(3, 1) contain
the point (1, 4). Find the acute angle of intersection of these lines to the
nearest degree.
14. The angle α, 0º α 180º, that a line makes with the positive x-axis is
called the angle of inclination of the line.
a. Find the angle of inclination of each of the following lines.
(ii) ៬r (6, 1) t(5, 1)
(i) ៬r (2, 6) t(3, 4)
b. Prove that the tangent of the angle of inclination is equal to the slope of
the line.
Application
246 C H A P T E R 7
15. You are driving from point A(24, 96) on a map grid toward point B with a
velocity defined by d៮៬ (85, 65) km/h.
a. State the parametric equations of the highway line.
b. How long have you been travelling when you reach a point P
102 km east of where you started at point A?
c. What are the coordinates of your position P at that time?
Thinking/Inquiry/
Problem Solving
16 a. By eliminating the parameter t from the parametric equations of a line,
xx
yy
show that the equation of a line can be written in the form a0 b0
(provided neither a nor b is zero). This is known as the symmetric
equation of a line.
b. Find a symmetric equation for each of the following lines.
(i) x 5 8t, y 3 5t
(ii) ៬r (0, 4) t(4, 1)
c. Find a symmetric equation for the line through the points A(7, 2) and
B(5, 4).
Part C
17. a. Show that P(5, 8) and Q(17, 22) are points on the line that passes
through A(7, 3) with direction vector (2, 5) .
b. Describe the line segment from P(5, 8) to Q(17, 22) using parametric
equations with suitable restrictions on the parameter.
Thinking/Inquiry/
Problem Solving
q are the position vectors of points P and Q in the plane.
18. a. Suppose ៮៬p and ៮៬
Show that the line that passes through P and Q has the vector equation
៮៬ tq៮៬.
r៬ (1 t)p
b. For what values of t does the point R with position vector r៬ lie between
points P and Q on the line?
c. When t 2, draw a vector diagram that shows where point R with position
vector r៬ lies on the line relative to points P and Q.
d. For what values of t does the point R with position vector r៬ lie closer to Q
than P?
19. a. Find the vector equations of the two lines that bisect the angles between the
lines
r៬1 (5, 2) t(3, 6)
៬r (5, 2) u(11, 2)
2
b. Sketch all four lines.
c. Are the two lines that bisect the angles made by the intersecting lines
always perpendicular? Explain.
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