The Vector Equation of a Plane

The Vector equation of a plane
To find the vector equation of a plane a point on the plane and two different direction vectors are required. The equation is defined as:

where a is the point on the plane and b and c are the vectors.

This equation can then be written as:

The Cartesian equation of a plane
The cartesian equation of the plane is easier to use. The equation is defined as:

One of the advantages to writing the equation in cartesian form is that we can easily find the normal (perpendicular) vector to the plane. It will be:

A proof of this is shown next.

The Cartesian equation of a plane and its normal
Look at the diagram. It shows a plane with the vector AB and a normal to the plane, n. AB is perpendicular to n, so:

So the vector n passes through these two points on the plane. This can be applied to any other point on the plane.

Finding the cartesian equation of the plane - 1
Method 1: Given a point on the plane and the plane’s normal. Find the cartesian equation of the plane containing the point (2,3,-1) and with normal vector .

Cartesian equation is ax+by+cz=d The normal to the plane is Equation is 5x-y+2z=d Substitute the point in to find d: 5(2)-(3)+2(-1)=5 .

5x-y+2z=5

Finding the cartesian equation of the plane - 2
Method 2: Given 3 points on a plane. Find the cartesian equation of the The cross product: plane containing the points A(2,3,1), B(4,0,5), and C(5,2,3).

Remember the work on the cross product: Find two vectors, in this case AB and AC, then calculate the cross product. This will give us the normal vector to the plane.

Normal vector to plane will be Plane has the equation 6x+10y+7z=d.

.

Substitute the point in to find d: -6(2)+10(3)+7(-1)=11

-6x+10y+7z=11