Perpendicular Line Calculator

Calculate the equation of a perpendicular line. Enter the equation of the original line and the point it passes through to calculate the perpendicular line equation.

Perpendicular Line Formula

Linear lines are almost always displayed in the form of

y = mx + b

Where m is the slope and b is the y intercept. The first step in finding the equation of a line perpendicular to another is understanding the relationship of their slopes. The slope of a perpendicular line is always the inverse to the other. This means that the product of the two slopes is equal to -1.

With that in mind we can formulate the following equations.

a * m = -1

a = -1 / m

Where m is the original slope, and a is the slope of the perpendicular line. Now that we have the slope of our new line, all we need now is the y-intercept, b.

To calculate the y-intercept, we can use a similar formula to the one used to calculate the equation of a parallel line.

b = y₀ + 1 * x₀ / m

  • Where b is the y intercept
  • y0 is the y coordinate the line passes through
  • X0 is the x coordinate the line passes through
  • m is the slope of the original line.

How to calculate a perpendicular line

Let’s look at an example of how to use these equations. First, lets assume you know the equation of the first line. It happens to be y=4x+5. Let’s also assume you know the x and y coordinates of a point that the perpendicular line passes through, say (4,5).

First, we need to calculate the slope. From the equation a = -1 / m we get a value of -1/4.

Next, we need to calculate the y intercept of the new line using the equation b = y₀ + 1 * x₀ / m . From this we get a value of 6.

Finally, we need to put this all together in the form: y=-1/4x + 6.

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