Perpendicular Lines and Slope

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

m_1 \times m_2 = -1

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 $-\frac{1}{4}$ . Are they perpendicular?

Multiply the slopes:

4 \times \left(-\dfrac{1}{4}\right)

= -\dfrac{4}{4} = -1

⟹ yes, the lines are perpendicular

Notice the two changes from 4 to $-\frac{1}{4}$ : 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.

y = mx + b

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

y - y_1 = m(x - x_1)

Use this when you know the slope and any point $(x_1,\ y_1)$ on the line, but not where it meets the $y$ -axis.

Example Building a perpendicular line

Find the equation of the line perpendicular to $y = 2x + 5$ that passes through $(4,\ 1)$ .

The given line has slope 2.

The perpendicular slope is the negative reciprocal:

-\dfrac{1}{2}

Check:

2 \times \left(-\dfrac{1}{2}\right) = -1

Use the point-slope form:

y - 1 = -\dfrac{1}{2}(x - 4)

⟹ y − 1 = −½(x − 4)

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

Note Parallel against perpendicular

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

y = mx + b

when you were given a point rather than the

y

-intercept.

Mistaking the point's

y

-value for

b

Losing a sign inside the bracket of the point-slope form when a coordinate is negative.

Summary

Two lines are perpendicular exactly when the product of their slopes is −1.
Equivalently, each slope is the negative reciprocal of the other: flip the fraction and change the sign.
Use y = mx + b when the slope and the y-intercept are known.
Use y − y₁ = m(x − x₁) when the slope and a point on the line are known.
Parallel means equal slopes; perpendicular means slopes multiplying to −1.