Let be the intersection of the sphere and the plane
- The intersection of any plane with any sphere is a circle. The plane in question passes through the centre of the sphere, so
has the same centre and same radius as the sphere. So has radius and centre - Notice that the point
satisfies both and and so is on We may choose to be the unit vector in the direction from the centre of the circle towards Namely - Since the plane of the circle is
the vector is perpendicular to the plane of So we may take - Then
Substituting in and gives
To check this, note that satisfies both and