]> 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 $( ( v ⟶ , v ⟶ ) − ( v ⟶ , w ⟶ ) * ( w ⟶ , v ⟶ ) ( w ⟶ , w ⟶ ) ) 1 / 2$ .