Here, we use our knowledge of the dot product to find the equation of a plane in R^{3} (3D space). Firstly, a **normal** vector to the plane is any vector that starts at a point in the plane and has a direction that is orthogonal (perpendicular) to the surface of the plane. For example, ** k** = (0,0,1) is a normal vector to the xy plane (the plane containing the x and y axes).

Any three distinct points define our plane, or alternatively, a single point in the plane and a given normal vector to the plane. We'll define it like this first.

All we need is a way of checking whether a given point, with position vector

Let

let

Then

(

Let

The two vectors,

both lie in the plane, so if we take their cross product we find our normal vector:

So as before our equation for the plane consists of all vectors,

(

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