The vector vec w→w is in the plane defined by vec u→u and vec v→v.
Unless vec u = -vec v→u=−→v, we have vec w ne 0→w≠0.
Let theta_uθu and theta_vθv be the angles the vector vec w→w makes with vec u→u and vec v→v, respectively. Then we have
vec w cdot vec u = |vec w| |vec u| cos theta_u →w⋅→u=∣∣→w∣∣∣∣→u∣∣cosθu
qquadqquad = (|vec u|vec v+|vec v| vec u)cdot vec u
qquad qquad = |vec u|(vec v cdot vec u)+|vec v| (vec u cdot vec u)
qquad qquad = |vec u|(vec v cdot vec u+|vec v| |vec u|) implies
|vec w| cos theta_u = vec v cdot vec u+|vec v| |vec u|
By interchanging vec u and vec v we find
|vec w| cos theta_v = vec u cdot vec v+|vec u| |vec v|
and so
|vec w| cos theta_u = |vec w| cos theta_v implies
color(red)(theta_u = theta_v)
Thus vec w bisects the angle between vec u and vec v (except in the special case vec u = -vec v )
