# 1.3 Parallel and perpendicular lines

We often wish to know whether two lines are *parallel* (that is, they never meet) or *perpendicular* (that is, they meet at right angles).

Two distinct lines, *y* = *m*_{1}*x* + *c*_{1} and *y* = *m*_{2}*x* + *c*_{2}, are parallel if and only if they have the same gradient; that is, if and only if *m*_{1} = *m*_{2}. For example, the lines *y* = −2*x* + 7 and *y* = −2*x* − 3 are parallel since both have gradient −2, whereas the lines *y* = −2*x* + 7 and *y* = 2*x* − 3 are not parallel since their gradients are not equal (they are −2 and 2, respectively).

Two lines ℓ_{1} and ℓ_{2} with equations *y* = *m*_{1}*x* + *c*_{1} and *y* = *m*_{2}*x* + *c*_{2}, respectively, where *m*_{1} and *m*_{2} are both non-zero, are perpendicular if and only if *m*_{1}*m*_{2} = −1. If the lines are perpendicular, then one (ℓ_{1}, say) must slope up from left to right and the other (ℓ_{2}, say) must slope down from left to right, as shown below.

Let the lines intersect at *P*, and let *Q* be a point on ℓ_{1} to the right of *P*. Suppose that *Q* is *a* units to the right of *P* and *b* units up from *P*, as illustrated above. Let *R* be the point on ℓ_{2} obtained by rotating *PQ* anticlockwise through an angle of /2; then *R* is *b* units to the left of *P* and *a* units up from *P*, as shown.

Then the gradient of ℓ_{1} is

and the gradient of ℓ_{2} is

It follows that

The proof of the converse is similar.

Two distinct lines with equations *y* = *m*_{1}*x* + *c*_{1} and *y* = *m*_{2}*x* + *c*_{2}, where *m*_{1} and *m*_{2} are both non-zero, are

**parallel**if and only if*m*_{1}=*m*_{2}and*c*_{1}≠*c*_{2};**perpendicular**if and only if*m*_{1}*m*_{2}= −1.

## Example 1

Determine which of the following lines are parallel, and which are perpendicular to each other.

### Answer

The gradients of the given lines are, respectively:

−2, , 2, , −2 and

Thus the lines ℓ_{1} and ℓ_{5} are parallel, and the lines ℓ_{2} and ℓ_{4} are perpendicular.

## Example 3

Determine which of the following lines are parallel, and which are perpendicular to each other.

### Answer

Since the gradient of a line whose equation is in the form *y* = *mx* + *c* is *m*, the gradients of the given lines are, respectively:

−2, 2, , , and −2

Thus ℓ_{1} and ℓ_{6} are parallel, and ℓ_{4} and ℓ_{5} are parallel; ℓ_{1} and ℓ_{4} are perpendicular, ℓ_{1} and ℓ_{5} are perpendicular, ℓ_{2} and ℓ_{3} are perpendicular, ℓ_{4} and ℓ_{6} are perpendicular, and ℓ_{5} and ℓ_{6} are perpendicular.