In this lesson, we’ll look at how to find the equation of a plane in cartesian and vector form.

## Equation of plane in vector form

To find the equation of a plane in vector form, you’ll first need to find a vector normal (i.e. perpendicular) to the plane. We call this vector normal to the plane the normal vector, **n**, of the plane.

Once n is found, we’ll still need a position vector of a point that lies on the plane. Assuming that **a** is the position vector of a point that lies on the plane, then the vector equation of the plane is: **r** . **n** = d, where **d**= **a.n**.

## Equation of plane in cartesian form

The cartesian equation of a plane takes the form ax + by + c = d, where a, b, c and d are constants to be determined.

## Convert the equation of a plane from vector form to cartesian form

To convert a plane equation from vector form to cartesian form involves rewriting it from **r** . **n** = d to ax + by + c = d.

To do so, we’ll rewrite **r** as:

Then do a dot product between **r** and **n**.

These will convert the vector equation of the plane to a cartesian equation.

## Convert the equation of a plane from cartesian form to vector form

To convert a plane equation from vector form to cartesian form involves rewriting it from ax + by + c = d to **r** . **n** = d.

Example:

Given the equation of the plane is 2x – 3y -z = 4, find (i) a vector normal to the plane (ii) find the vector equation of the plane.

## Notes on H2 Math Vectors

You’ll find all the notes on H2 A Level Math Vectors topic here.

## All the notes for H2 A Level Math

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