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Parallel and perpendicular lines

Parallel and perpendicular lines

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Explainer Video

Tutor: Toby

Summary

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 yyy-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 mmm, then almost any other line with gradient mmm 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 yyy-intercepts are different, the lines are parallel. As equations, a line parallel to y=mx+cy=mx+cy=mx+c is given by the equation

y=mx+dy=mx+dy=mx+d


where ddd is not equal to ccc.​


Example 1

Find a line that is parallel to 

y=−4x+7y=-4x+7y=−4x+7


First look for the gradient of the line. It is −4-4−4, thus any parallel line will also have this gradient. The yyy-intercept of this line is 777, so you must pick any other ccc-term for your parallel line. For example 555. Now you have that a parallel line to y=−4x+7y=-4x+7y=−4x+7​ is

y=−4x+5‾\underline{y=-4x+5}y=−4x+5​


This is just one example. There are infinitely more parallel lines each with a different yyy-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 yyy-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+cy=mx+cy=mx+c, identify the gradient of your line. This is mmm.​
2.
The reciprocal of a number is 111 divided by that number. Find the reciprocal of mmm. This is 1m\frac1mm1​.​
3.
Find the negative of the reciprocal. This is −1m-\frac1m−m1​. This is the gradient of the perpendicular line.​

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

y=−1mx+dy=-\frac1mx+dy=−m1​x+d


where ddd​ 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 111 divided by that number. Hence the reciprocal of −3-3−3 is 1−3\frac1{-3}−31​ which is the same as −13-\frac13−31​. You are looking for the negative reciprocal, so you want the negative of −13-\frac13−31​. This is 

−(−13)=13‾-(-\frac13)=\underline{\frac13}−(−31​)=31​​​​


Example 3

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


Start by working out the gradient of the perpendicular line. This will be the negative reciprocal of the gradient of y=8x+2y=8x+2y=8x+2. The gradient of this line is 888, so the negative reciprocal is −18-\frac18−81​. You can now pick any yyy-intercept, for example -3. Thus the equation of a perpendicular line to y=8x+2y=8x+2y=8x+2 is 

y=18x−3‾\underline{y=\frac18x-3}y=81​x−3​


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


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FAQs - Frequently Asked Questions

Can a line be parallel to itself?

A line is not parallel with itself since it is itself.

What is the negative reciprocal of a number?

The negative reciprocal of a number is -1 divided by that number.

What does it mean for lines to be perpendicular?

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.

What does it mean for lines to be parallel?

If two lines are parallel, they never intersect and run in the same direction forever. More technically, they have the same gradient.

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