Perpendicular Lines and Slope

Two lines are perpendicular exactly when their slopes multiply to -1. The negative reciprocal shortcut, plus the two standard forms for writing the equation of a line.

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Parallel lines share the same slope. Perpendicular lines have a different relationship, and it is just as easy to test: multiply the two slopes together and see whether you get −1.

Concept The condition

If two lines are perpendicular, the product of their slopes is −1. The converse holds too: if the product is −1, the lines must be perpendicular.

The same relationship can be phrased differently: each slope is the negative reciprocal of the other. Flip the fraction and change the sign.

Example Testing for perpendicularity

One line has slope 4 and another has slope . Are they perpendicular?

Multiply the slopes: .
.
⟹ yes, the lines are perpendicular

Notice the two changes from 4 to : the fraction was inverted and the sign was flipped. Doing only one of the two would not give a perpendicular line.

Concept Two ways to write a line's equation

Which form you use depends on what you are given.

Use this when you know the slope and the point where the line crosses the -axis, .

Use this when you know the slope and any point on the line, but not where it meets the -axis.

Example Building a perpendicular line

Find the equation of the line perpendicular to that passes through .

The given line has slope 2.
The perpendicular slope is the negative reciprocal: .
Check: .
Use the point-slope form: .
⟹ y − 1 = −½(x − 4)

The point-slope form was the right choice here because a point was given rather than a -intercept.

Note Parallel against perpendicular
equal slopes product = −1
Parallel — the slopes are equal.
Perpendicular — the slopes multiply to −1.
Note Mistakes to avoid
Inverting the fraction but forgetting to change the sign — both steps are needed.
Expecting perpendicular slopes to be equal; that is the parallel condition.
Using when you were given a point rather than the -intercept.
Mistaking the point's -value for .
Losing a sign inside the bracket of the point-slope form when a coordinate is negative.
Summary
  1. Two lines are perpendicular exactly when the product of their slopes is −1.
  2. Equivalently, each slope is the negative reciprocal of the other: flip the fraction and change the sign.
  3. Use y = mx + b when the slope and the y-intercept are known.
  4. Use y − y₁ = m(x − x₁) when the slope and a point on the line are known.
  5. Parallel means equal slopes; perpendicular means slopes multiplying to −1.
283. Relations and Functions
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