### 1 Perpendicular lines

One form for the equation of a straight line is

$\phantom{\rule{2em}{0ex}}y=mx+c$

where $m$ and $c$ are constants. We remember that $m$ is the gradient of the line and its value is the tangent of the angle $\theta$ that the line makes with the positive $x$ -axis. The constant $c$ is the value obtained where the line intersects the $y$ -axis. See Figure 1:

Figure 1

If we have a second line, with equation

$\phantom{\rule{2em}{0ex}}y=nx+d$

then, unless $m=n$ , the two lines will intersect at one point. These are drawn together in Figure 2. The second line makes an angle $\psi$ with the positive $x$ -axis.

Figure 2

A simple question to ask is “what is the relation between $m$ and $n$ if the lines are perpendicular?” If the lines are perpendicular, as shown in Figure 3, the angles $\theta$ and $\psi$ must satisfy the relation:

$\phantom{\rule{2em}{0ex}}\psi -\theta =9{0}^{\circ }$

Figure 3

This is true since the angles in a triangle add up to $18{0}^{\circ }$ . According to the figure the three angles are $9{0}^{\circ }$ , $\theta$ and $18{0}^{\circ }-\psi$ . Therefore

$\phantom{\rule{2em}{0ex}}18{0}^{\circ }=9{0}^{\circ }+\theta +\left(18{0}^{\circ }-\psi \right)\phantom{\rule{2em}{0ex}}\text{implying}\phantom{\rule{2em}{0ex}}\psi -\theta =9{0}^{\circ }$

In this special case that the lines are perpendicular or normal to each other the relation between the gradients $m$ and $n$ is easily obtained. In this deduction we use the following basic trigonometric relations and identities:

$\phantom{\rule{2em}{0ex}}sin\left(A-B\right)\equiv sinAcosB-cosAsinB\phantom{\rule{2em}{0ex}}cos\left(A-B\right)\equiv cosAcosB+sinAsinB$

$\phantom{\rule{2em}{0ex}}tanA\equiv \frac{sinA}{cosA}\phantom{\rule{2em}{0ex}}sin9{0}^{\circ }=1\phantom{\rule{2em}{0ex}}cos9{0}^{\circ }=0$

Now

##### Key Point 1

Two straight lines $y=mx+c$ , $y=nx+d$ are perpendicular if

This result assumes that neither of the lines are parallel to the $x$ -axis or to the $y$ -axis, as in such cases one gradient will be zero and the other infinite.
##### Exercise

Which of the following pairs of lines are perpendicular?

1. $y=-x+1,\phantom{\rule{1em}{0ex}}y=x+1$
2. $y+x-1=0,\phantom{\rule{1em}{0ex}}y+x-2=0$
3. $2y=8x+3,\phantom{\rule{1em}{0ex}}y=-0.25x-1$
1. perpendicular
2. not perpendicular
3. perpendicular