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Both of these vectors can be multiplied by 4 to ensure that both direction numbers
are integers. As a result, either m ! 11, 142 or m ! 11, 142 are the best choices
for a direction vector. When we determine the direction vector, any scalar multiple of
this vector of the form t11, 142 is correct, provided that t 0. If t 0, 10, 02 would
be the direction vector, meaning that the line would not have a defined direction.
Expressing the Equations of Lines Using Vectors
In general, we would like to determine the equation of a line if we have a
direction for the line and a point on it. In the following diagram, the given point
!
P 0 1x 0 , y 0 2 is on the line L and is associated with vector OP
!0, designated as
The direction of the line is given by where is any vector
collinear with m ! m ! tm ! r 0 .
1a, b2, , tR
. P1x, y2 represents a general point on the line, where OP !
is the
vector associated with this point.
O
y
r = r 0 + tm, t P R
L
P tm
0 (x 0 , y 0 ) P(x, y)
r r 0
m = (a, b)
To find the vector equation of line L, the triangle law of addition is used.
In OP ! !
^OP OP 0 P0 P ! 0 P,
.
x
! ! ! !
Since and the vector equation of the line is
written as r ! tm ! P
! ! 0 P ! r OP , r 0 OP0 ,
,
r 0 tm , tR.
When writing an equation of a line using vectors, the vector form of the line is
sometimes modified and put in parametric form. The parametric equations of a
line come directly from its vector equation. How to change the equation of a line
from vector to parametric form is shown below.
The general vector equation of a line is r ! ! !
r 0 tm , tR.
In component form, this is written as 1x, y2 1x 0 , y 0 2 t1a, b2, tR. Expanding
the right side, 1x, y2 1x 0 , y 0 2 1ta, tb2 1x 0 ta, y 0 tb2, tR. If we equate
the respective x and y components, the required parametric form is x x 0 ta
and y y 0 tb, tR.
Vector and Parametric Equations of a Line in R 2
Vector Equation: r ! ! !
r 0 tm , tR
Parametric Equations: x x
!
0 ta, y y 0 tb, tR
where r is the vector from 10, 02 to the point 1x and m !
0 0 , y 0 2 is a direction
vector with components 1a, b2.
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8.1 VECTOR AND PARAMETRIC EQUATIONS OF A LINE IN R 2
NEL