Consider a plane with the Cartesian equation

Find each of the following:

- A normal vector to the plane with length 1.
- The intercepts of the plane.
- The distance from the plane to the origin.
- The point on the plane, , nearest to the origin.

- From the Cartesian equation we know that a normal vector to the plane is . Making this of unit length, we have a unit normal given by
- Since the Cartesian equation gives us we have the -intercept is , the -intercept is and the -intercept is .
- The distance from the plane to the origin is given by
- The point on the plane which is nearest to the origin is in the direction of the normal vector, and at a distance of , therefore,

d) is incorrect here and correct in the book. The catch is that (PN) is negative, so the normal vector point in the opposite direction from the vector P on the plane, so we should negate it’s scaled version in order for that point to be on the plane.