Let x and y both be vectors in 2 or 3space and let a y and

Let x and y both be vectors in 2- or 3-space, and let a = ||y|| and b = ||x||. Prove that the vector v = ax + by bisects the angle between x and y.

v.x = a|x|^2 + bx.y = |y||x|^2 + |x|x.y

v.x/|x||v| = (|x||y| +x.y)/|v|

v.y = ax.y + b|y|^2 = |y|x.y + |x||y|^2

v.y/|y||v| = (x.y+|x||y|)/|v|

v.x/|x||v| = v.y/|v||y|

angle between v,x and v,y are equal

v bisects angle between x and y

thus proved