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Express the component of v perpendicular to w in terms of dot products.

Solution:

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 of w.
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