Equations of Planes

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The Vector equation of a planeTo 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 planeThe 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 normalLook 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 - 1Method 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 - 2Method 2: Given 3 points on a

plane.Find the cartesian equation of the 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.

The cross product:

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