Proof that Vector v Bisects the Angle Created by Vectors a and b

[meson0731]

Homework Statement

Given vector a and b, let vector v = (|a|b + |b|a) / (|a| + |b|). Show that vector v bisects the angle created by vector b and a.

Homework Equations

cos(θ) = a dot b / (|a||b|)

The Attempt at a Solution

I used the dot product equation to find angle between a and v.

cosθ = (a dot v) / |a||v|

i substituted ((|a|b + |b|a) / (|a| + |b|)) for v.

cosθ = (a dot ((|a|b + |b|a) / (|a| + |b|)) / |a||v|

I distributed the (a dot ((|a|b + |b|a)

cosθ ( ((|a|b dot a) + (|b| |a|^2) ) / (|a| + |b|)) / (|a||b|)

I then tried to get the cosθ of b dot w to the same thing. I tried to get the two angles equal to each other. My idea is that if the angles formed by a and v and b and v are equal then v bisects the two angles.

 

[vela]

So you have
$$\cos\theta_{av} = \frac{a(\vec{a}\cdot\vec{b})+a^2b}{(a+b)av}$$ where $a=|\vec{a}|$, $b=|\vec{b}|$, and $v=|\vec{v}|$, and the corresponding expression for $\cos\theta_{bv}$. So far so good. Try calculating
$$\frac{\cos\theta_{av}}{\cos\theta_{bv}}$$ and show it's equal to 1.

 

[Ratch]

Meson,

The problem becomes easy when you realize that if you add two different vectors of equal lengths, then you get a third vector whose direction bisects the angle of the first two vectors. Try it and see. That takes care of the geometry. Now see if you can figure out the algebra.

Ratch