# Equations of Perpendicular Lines

## Key Questions

If there is no specific point that the perpendicular line must pass through, all lines with slopes that are opposite reciprocals of the original line are perpendicular to it.

#### Explanation:

For example, the line $y = 3 x$ has a slope of 3.

Any line that is perpendicular to it must be opposite (negative instead of positive or vice versa) reciprocals (multiplicative inverse, reciprocals multiply to 1, flip the numerator and denominator)

Opposite of 3: -3

Reciprocal of -3 ($- \frac{3}{1}$): $- \frac{1}{3}$

Lines that are perpendicular to $y = 3 x$ must have a slope of $- \frac{1}{3}$. That is the only requirement. The y-intercepts can vary.

Perpendicular lines to $y = 3 x$:

$y = - \frac{1}{3} x + 4$

$y = - \frac{1}{3} x - 12$

$y = - \frac{1}{3} x$

The y-intercepts change where the lines intersect, but all these lines are perpendicular to $y = 3 x$.

• Say we have the two lines:
$\text{Line 1} : y = {m}_{1} x + {c}_{1}$
$\text{Line 2} : y = {m}_{2} x + {c}_{2}$

If ${m}_{1} = {m}_{2}$ then the lines are parallel, since they both have the same rate of change and therefore will never cross.

If ${m}_{1} {m}_{2} = - 1$ then they are perpendicular.

• The slopes of perpendicular lines are negative reciprocal of each other.

I hope that this was helpful.