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 of 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, let’s 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.
Finding Parallel and Perpendicular Lines
in the 2-Dimension Cartesian Plane, all straight lines can be represented as an equation of the form y=mx+b, where m is the slope and x and y are points along the line. Since all lines can be described this way, it makes calculating parallel and perpendicular lines simple.
It’s just a matter of manipulating the equation. Let’s look at how the equation is manipulated in order to calculate the equation of a perpendicular line.
Conceptually, a perpendicular line is a line that crosses through the original line at any point and forms a 90-degree angle at intercept. Since there can be an infinite number of perpendicular lines, calculating any specific line requires a point.
The slope of a perpendicular line is the reciprocal of the slope of the original line. This simply means if the original slope is m, the reciprocal is 1/m. With this in mind, we can now manipulate the equation to determine the perpendicular line.
y=mx+b —> y2=(1/m)x2+b
The only thing left to do is solve for be using the point given along the perpendicular line.
Steps to calculate a perpendicular line
We’ve gone over how to calculate the equation of a perpendicular line, and how manipulation of the original equation yields the results, but the following steps will outline the specific steps.
- Calculate the slope of the original line. This can be done through the use of two points along the original line. To learn more about calculating the slope of a line click here.
- Take the reciprocal of the original slope. If the slope is m, the reciprocal is 1/m.
- Calculate b, the y-intercept of the new line using the new slope and the point given along that line.
- Put all of the information together in point-slope form.
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