# Parallel and perpendicular lines

## In a nutshell

Given the equation of a line, the equations for parallel and perpendicular lines can be found. There are infinitely many such lines, since only the gradient matters - the $y$-intercept can be essentially anything.

## Parallel lines

If two lines are parallel, they never intersect and run in the same direction forever. More technically, they have the same gradient. Hence if a line has gradient $m$, then almost any other line with gradient $m$ is parallel. The only exception is the line itself: a line is not parallel with itself since it *is* itself. Thus, if the gradients of two lines are the same and their $y$-intercepts are different, the lines are parallel. As equations, a line parallel to $y=mx+c$ is given by the equation

$y=mx+d$

where $d$ is not equal to $c$.

##### Example 1

*Find a line that is parallel to *

*$y=-4x+7$*

*First look for the gradient of the line. It is $-4$, thus any parallel line will also have this gradient. The $y$-intercept of this line is $7$, so you must pick any other $c$-term for your parallel line. For example $5$. Now you have that a parallel line to $y=-4x+7$ is*

*$\underline{y=-4x+5}$*

*This is just one example. There are infinitely more parallel lines each with a different $y$-intercept.*

## Perpendicular lines - higher only

If two lines are perpendicular to each other, they are at right-angles to each other. This means there is a right-angle at their intersection. Again, you use the gradient of one line to find the gradient of the other. There are infinitely many perpendicular lines to a given line since the $y$-intercept can be any number.

To find the gradient of the perpendicular line, you find the negative reciprocal of the gradient of the original line. To find the negative reciprocal of a gradient, follow this procedure:

#### procedure

**1.**
| Given a line with equation $y=mx+c$, identify the gradient of your line. This is $m$. |

**2.**
| The reciprocal of a number is $1$ divided by that number. Find the reciprocal of $m$. This is $\frac1m$. |

**3.**
| Find the negative of the reciprocal. This is $-\frac1m$. This is the gradient of the perpendicular line. |

Hence the equation of a line perpendicular to $y=mx+c$ is given by

$y=-\frac1mx+d$

where $d$ is any constant number.

##### Example 2

*What is the negative reciprocal of -3?*

*From the procedure above, you see that the reciprocal of a number is $1$ divided by that number. Hence the reciprocal of $-3$ is $\frac1{-3}$ which is the same as $-\frac13$. You are looking for the negative reciprocal, so you want the negative of $-\frac13$. This is *

*$-(-\frac13)=\underline{\frac13}$*

##### Example 3

*Find a line that is perpendicular to the line $y=8x+2$.*

*Start by working out the gradient of the perpendicular line. This will be the negative reciprocal of the gradient of $y=8x+2$. The gradient of this line is $8$, so the negative reciprocal is $-\frac18$. You can now pick any $y$-intercept, for example -3. Thus the equation of a perpendicular line to $y=8x+2$ is *

*$\underline{y=\frac18x-3}$*

*As with parallel lines, this is just one of infinitely many answers to this problem.*