W= |u| * v +|v| * u Where u and v and w are all non-zero vectors Show that w bisects the angle between u and v ?

1 Answer
May 25, 2018

See below:

Explanation:

The vector #vec w# is in the plane defined by #vec u# and #vec v#.

Unless #vec u = -vec v#, we have #vec w ne 0#.

Let #theta_u# and #theta_v# be the angles the vector #vec w# makes with #vec u# and #vec v#, respectively. Then we have

#vec w cdot vec u = |vec w| |vec u| cos theta_u #
#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# )