Perpendicular Lines

Idea

Turn a line by a right angle and its rise becomes its run.

A line that climbs $3$ for every $1$ across becomes a line that falls $1$ for every $3$ across. The numbers swap places and the sign changes. That is why the two slopes multiply to $-1$ .

Rule

Two lines with slopes $m_1$ and $m_2$ are perpendicular exactly when

m_1 \cdot m_2 = -1

Equivalently, each slope is the negative reciprocal of the other:

m_2 = -\frac{1}{m_1}

In standard form the condition is $a_1a_2 + b_1b_2 = 0$ , which has the advantage of surviving the vertical case.

How it is used

Building a perpendicular line is like building a parallel one, but you change the slope first.

Find the original slope $m_1$ .
Flip and negate it to get $m_2 = -\dfrac{1}{m_1}$ .
Use the point with the point-slope form.

Original slope	Perpendicular slope
$2$	$-\dfrac{1}{2}$
$-\dfrac{3}{4}$	$\dfrac{4}{3}$
$1$	$-1$
$0$ (horizontal)	undefined (vertical)

The last row is a case the multiplication rule cannot handle.

A horizontal line and a vertical line are perpendicular. But one slope is $0$ and the other does not exist, and $0 \times \text{undefined}$ has no meaning. For this pair, either use the picture, or use $a_1a_2 + b_1b_2 = 0$ , which works for every case.

There is a shortcut here too. A line perpendicular to $ax + by + c = 0$ has the form $bx - ay + k = 0$ . Swap the two coefficients and change the sign of one.

Worked example

Find the line through $(4, 1)$ perpendicular to $2x + y - 3 = 0$ . Rearranged, the original is $y = -2x + 3$ , so $m_1 = -2$ . The perpendicular slope is the negative reciprocal:

m_2 = -\frac{1}{-2} = \frac{1}{2}

Now point-slope through $(4, 1)$ : $y - 1 = \dfrac{1}{2}(x - 4)$ . Multiply by $2$ to clear the fraction: $2y - 2 = x - 4$ , and collect:

x - 2y - 2 = 0

Explore

Drag C and D until the verdict says Perpendicular. Read the two slopes and multiply them. You always get $-1$ . Now make the blue line horizontal and try again. The only perpendicular line is vertical, and its slope reads undefined.

Two lines, one plane

Try this. Use A(0,0), B(1,1), C(0,1), D(1,2) for parallel lines. Change D to (1,0) for perpendicular lines.

Point A · x-2Point A · y3Point B · x3Point B · y1Point C · x-3Point C · y-2Point D · x3Point D · y4

Blue slope: -0.4 · Green slope: 1 · Intersecting at (0.857, 1.857).

Idea

Rule

How it is used

Worked example

Explore

Try it yourself.

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