Given the equation of a plane: ax+by+cz=D, or in vector notation , how do we find the (shortest) distance of a plane to the origin?

(When a question asks for the distance of a plane to the origin, by definition it means the shortest distance.)

One way to derive the formula is this:

## Derivation

Let X be the point on the plane nearest to the origin.

must be perpendicular to the plane, i.e. parallel to the normal vector .

Furthermore, X lies on the plane, hence we have

Using the formula for dot product, we can get

Since is parallel to , is either 0 or 180 degrees, hence is either 1 or -1.

Thus, we have .

The shortest distance from the point X to the origin is then

**Ans**: Shortest distance from point to plane is

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