Part A
1. What is a direction vector? What is a parameter? What role do these quanti- ties play in the equation of a line?
2. State a direction vector for each of the following lines.
a. a line parallel to x93t,y 4t b. the line through (6, 4) and (2,6) c. the line y3x6
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. x38t,y4t b. r(4, 0)t(0, 5)
4. State a parametric equation and a vector equation for each of the following lines.
a. b.
5. Graph the following lines.
a. x 15t b. r(3,4)t(4, 3)
y62t
6. For each of the following, find the parametric equations of the line that passes through the pointPwith direction vector d. In each case, find two points on the line different from P.
a. P(1, 1),d(4,4) b. P(5, 0),d(1, 3) 7. State parametric equations
a. for the x-axis
b. for a line parallel to but not coincident with the x-axis
x y
x y
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7 . 1 PA R A M E T R I C A N D V E C TO R E Q U AT I O N S O F A L I N E I N A P L A N E 245
Part B
8. For each of the following lines, find the vector equation that passes through the point Pwith direction vector d.
a. P(2, 7),d(3,4) b. P冢2,34冣,d冢23, 6冣
c. P(1,1),d(兹3苶, 3) d. P(0, 0),d(2, 3)
9. For each of the following lines, state a direction vector with integer compo- nents. If possible, name a point on the line with integer coordinates.
a. x1
32t,y32 3t b. r冢13,12冣t冢13,14冣
c. r冢12, 3冣t冢12, 5冣
10. For each of the following, determine which pairs of lines are parallel and which are perpendicular.
a. x13t,y74t and x24s,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. x6,y17t b. r(5, 10)t(1, 5) c. r(2, 3)t(3,1)
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, 0ºα180º, that a line makes with the positive x-axis is called the angle of inclinationof the line.
a. Find the angle of inclination of each of the following lines.
(i) r(2,6)t(3,4) (ii) r(6, 1)t(5, 1)
b. Prove that the tangent of the angle of inclination is equal to the slope of the line.
15. You are driving from point A(24, 96) on a map grid toward point Bwith a velocity defined by d(85,65) km/h.
Application Application Thinking/Inquiry/
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C H A P T E R 7 246
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 Pat that time?
16 a. By eliminating the parameter tfrom the parametric equations of a line, show that the equation of a line can be written in the form xax0y
b y0
(provided neither anor bis zero). This is known as the symmetric equationof a line.
b. Find a symmetric equation for each of the following lines.
(i) x58t,y 35t (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.
18. a. Suppose pand qare the position vectors of points Pand Qin the plane.
Show that the line that passes through Pand Qhas the vector equation r(1t)ptq.
b. For what values of tdoes the point Rwith position vector rlie between points Pand Qon the line?
c. When t2, draw a vector diagram that shows where point Rwith position vector rlies on the line relative to points Pand Q.
d. For what values of tdoes the point Rwith position vector rlie closer to Q than P?
19. a. Find the vector equations of the two lines that bisect the angles between the lines
r1(5, 2)t(3, 6) r2(5, 2)u(11, 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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7 . 1 PA R A M E T R I C A N D V E C TO R E Q U AT I O N S O F A L I N E I N A P L A N E 247
