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In either vector or parametric form, t is called a parameter. This means that t can be
replaced by any real number to obtain the coordinates of points on the line.
EXAMPLE 2
Reasoning about the vector and parametric equations of a line
a. Determine the vector and parametric equations of a line passing through point
A11, 42 with direction vector m ! 13, 32.
b. Sketch the line, and determine the coordinates of four points on the line.
c. Is either point Q121, 232 or point R129, 342 on this line?
Solution
a. Since A(1, 4) is on the line, and The
vector equation is r ! OP ! m ! 0 r ! 0 11, 42 13, 32.
11, 42 t13, 32, tR. The parametric equations are
x 1 3t, y 4 3t, tR.
It is also possible to use other scalar multiples of m ! 13, 32 as a direction
vector, such as which gives the respective vector and parametric
equations r ! 11, 12,
11, 42 s11, 12, sR, and x 1 s, y 4 s, sR. The
vector 11, 12 has been chosen as our direction vector for the sake of
simplicity. Note that we have written the second equation with parameter
s to avoid confusion between the two lines. Although the two equations,
r ! 11, 42 t13, 32, tR, and r ! 11, 42 s11, 12, sR, appear with
different parameters, the lines they represent are identical.
b. To determine the coordinates of points on the line, the parametric equations
x 1 s, y 4 s, sR, were used, with s chosen to be 0, 1, 1, and 6.
To find the coordinates of a particular point, such as D, s 6 was substituted
into the parametric equations and x 1 162 7, y 4 162 2.
The required point is D17, 22. The coordinates of the other points are
determined in the same way, using the other values of s.
y
6
s = 1, A(0, 5)
4 s = 0, B(1, 4)
s = –1, C(2, 3)
2
m = (– 1, 1)
x
–2 0 2 4 6 8
–2
s = –6, D(7, –2)
–4
–6
NEL
CHAPTER 8
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