]> Exercise 3.4

Exercise 3.4

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 ( v , w ) ( w , w ) w .
The projection of v normal to w is then v ( v , w ) ( w , w ) w , the component of v normal to w is then the square root of the dot product of this projection with itself, or ( ( v , v ) ( v , w ) * ( w , v ) ( w , w ) ) 1 / 2 .