Express the component of v perpendicular to w in terms of dot products.
The component of v perpendicular to w is the magnitude of the
projection of v perpendicular to w. By definition
this projection is v minus the projection of v in the direction
But the projection of v in the direction of w is
The projection of v normal to w is then , the component of v normal to w is then the square root of the dot product of this projection with itself, or