1 Perpendicular lines

One form for the equation of a straight line is

$y=mx+c$

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

$y=nx+d$

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:

$\psi -\theta =90^{\circ }$

Figure 3

This is true since the angles in a triangle add up to $180^{\circ }$ . According to the figure the three angles are $90^{\circ }$ , $\theta$ and $180^{\circ }-\psi$ . Therefore

$180^{\circ }=90^{\circ }+\theta +\left(180^{\circ }-\psi \right)\text{implying}\psi -\theta =90^{\circ }$

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:

$sin\left(A-B\right)\equiv sinAcosB-cosAsinBcos\left(A-B\right)\equiv cosAcosB+sinAsinB$

$tanA\equiv \frac{sinA}{cosA}sin90^{\circ }=1cos90^{\circ }=0$

Now

Exercise

Which of the following pairs of lines are perpendicular?

1. $y=-x+1,y=x+1$
2. $y+x-1=0,y+x-2=0$
3. $2y=8x+3,y=-0.25x-1$

Answer

1. perpendicular
2. not perpendicular
3. perpendicular