One form for the equation of a straight line is
where and are constants. We remember that is the gradient of the line and its value is the tangent of the angle that the line makes with the positive -axis. The constant is the value obtained where the line intersects the -axis. See Figure 1:
If we have a second line, with equation
then, unless , the two lines will intersect at one point. These are drawn together in Figure 2. The second line makes an angle with the positive -axis.
A simple question to ask is “what is the relation between and if the lines are perpendicular?” If the lines are perpendicular, as shown in Figure 3, the angles and must satisfy the relation:
This is true since the angles in a triangle add up to . According to the figure the three angles are , and . Therefore
In this special case that the lines are perpendicular or normal to each other the relation between the gradients and is easily obtained. In this deduction we use the following basic trigonometric relations and identities:
Two straight lines , are perpendicular if
Which of the following pairs of lines are perpendicular?
- not perpendicular