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It is not possible, in this case, to calculate the slope because the line has direction
2
vector meaning its slope would be which is undefined. Since the line is
0 ,
parallel to the y-axis, it must have the form x a, where 1a, 02 is the point where
the line crosses the x-axis. The equation of this line is x 1 0 or x 1.
10, 22, NEL
Developing the Cartesian Equation from a Direction Vector
In addition to making the connection between lines in either slope–y-intercept form
or Cartesian form with those in vector form, we would like to consider how direction
vectors can be used to obtain the equations of lines in Cartesian form.
In the following diagram, the line L represents a general line in R 2 . A line has
been drawn from the origin, perpendicular to L. This perpendicular line is called
the normal axis for the line and is the only line that can be drawn from the origin
perpendicular to the given line. If the origin is joined to any point on the normal
axis, other than itself, the vector formed is described as a normal to the given line.
Since there are an infinite number of points on the normal axis, this is a way of
saying that any line in R 2 has an infinite number of normals, none of which is the
zero vector. A general point on the normal axis is given the coordinates
and so a normal vector, denoted by n ! , is the vector n ! N1A, B2,
1A, B2.
normal axis
y
n = (A, B)
L
O(0,0)
x
The important property of the normal vector is that it is perpendicular to any
vector on the given line. This property of normal vectors is what allows us to
derive the Cartesian equation of the line.
In the following diagram, the line L is drawn, along with a normal n ! 1A, B2,
to L. The point P1x, y2 represents any point on the line, and the point P 0 1x 0 , y 0 2
represents a given point on the line.
normal axis
y
n = (A, B)
L
P 0 (x 0 , y 0 )
P (x, y)
m = (x – x 0 , y – y 0 )
x
O(0,0)
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8.2 CARTESIAN EQUATION OF A LINE