A plane can be expressed through parametrization or an equation. The equation of a plane through a point with a normal vector looks like this:
Formula
This can also be written as
You can find the parametric equation of a plane by using two vectors and a point. Given that and spans the plane and is a point in the plane, the parametric equation can be written like this:
Theory
You can find both the equation and the parametric equation of a plane if you know three points in the plane. With three points , and you can find and , which you can use to find a parametric equation for the plane. You can also use the cross product of these vectors to find a normal vector to the plane, which then can be used to find the equation of the plane.
Rule
The equation of the -plane is
The equation of the -plane is
The equation of the -plane is
Example 1
If a plane has the equation , you can find a normal vector by looking at the numbers in front of the variables, giving you .
Example 2
The points , and are in a plane . Describe the plane with an equation and a parametric equation.
First, you find the vectors and :
To find the parametric equation, you insert the point and the vectors and into the formula:
Example 3
A point is in a plane with a normal vector . What is the equation and the parametric equation of the plane?
You can find the equation of the plane first. You already have everything you need, so you can just insert the coordinates of the point and the values in the normal vector into the general equation of a plane:
You can find another two points in the plane by using the equation of the plane. If you set two of the coordinates to , you will find the third coordinate very quickly. For example, you can set to find :
This means that is a point in the plane. You can do the same with to find :
Then, is a point in the plane as well. You now have three points in the plane, which can be used to create two vectors! That will give you