Exercise 8.2
PART A
1. A line has y 56 x 9 as its equation.
a.
b.
c.
d.
e.
2. a.
b.
c.
Give a direction vector for a line that is parallel to this line.
Give a direction vector for a line that is perpendicular to this line.
Give the coordinates of a point on the given line.
In both vector and parametric form, give the equations of the line parallel
to the given line and passing through A17, 9 2.
In both vector and parametric form, give the equations of the line
perpendicular to the given line and passing through B12, 12.
!
Sketch the line defined by the equation r 12, 12 s12, 5 2, s苸R.
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On the same axes, sketch the line q 12, 5 2 t12, 1 2 , t苸R.
Discuss the impact of switching the components of the direction vector
with the coordinates of the point on the line in the vector equation of a line
in R2.
3. For each of the given lines, determine the vector and parametric equations.
7
a. y x 6
8
C
3
b. y x 5
2
c. y 1
d. x 4
4. Explain how you can show that the lines with equations x 3y 4 0 and
6x 18y 24 0 are coincident.
5. Two lines have equations 2x 3y 6 0 and 4x 6y k 0.
a. Explain, with the use of normal vectors, why these lines are parallel.
b. For what value of k will these lines be coincident?
PART B
6. Determine the Cartesian equation for the line with a normal vector of 14, 5 2,
passing through the point A11, 52.
7. A line passes through the points A13, 5 2 and B12, 4 2. Determine the
Cartesian equation of this line.
8. A line is perpendicular to the line 2x 4y 7 0 and that passes through
the point P17, 2 2. Determine the equation of this line in Cartesian form.
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9. A line has parametric equations x 3 t, y 2 4t, t苸R.
a. Sketch this line.
b. Determine a Cartesian equation for this line.
CHAPTER 8
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A
10. For each pair of lines, determine the size of the acute angle, to the nearest
degree, that is created by the intersection of the lines.
a. 1x, y2 13, 62 t12, 5 2 and 1x, y2 13, 4 2 t14, 1 2
b. x 2 5t, y 3 4t and x 1 t, y 2 6t
c. y 0.5x 6 and y 0.75x 1
d. 1x, y2 11, 1 2 t12, 4 2 and 2x 4y 8
e. x 2t, y 1 5t and 1x, y2 14, 0 2 t14, 12
f. x 3 and 5x 10y 20 0
11. The angle between any pair of lines in Cartesian form is also the angle
between their normal vectors. For the lines x 3y 6 0 and
x 2y 7 0, do the following:
a. Sketch the lines.
b. Determine the acute and obtuse angles between these two lines.
T
12. The line segment joining A13, 2 2 and B18, 4 2 is the hypotenuse of a right
triangle. The third vertex, C, lies on the line with the vector equation
1x, y2 16, 62 t13, 42 .
a. Determine the coordinates of C.
b. Illustrate with a diagram.
c. Use vectors to show that ⬔ACB 90°.
PART C
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!
13. Lines L1 and L2 have n1 and n2 as their respective normals. Prove that the
angle between the two lines is the same as the angle between the two normals.
y
B
n1
u
C
n2
A
L2
O
x
L1
(Hint: Show that ⬔AOC u by using the fact that the sum of the angles in a
quadrilateral is 360°.)
14. The lines x y 1 0 and x ky 3 0 have an angle of 60° between
them. For what values of k is this true?
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