# Bisector of two vectors

Given two vectors $$\mathbf{a}$$ and $$\mathbf{b}$$, a bisector vector $$\mathbf{c}$$ can be determined by

$\mathbf{c} = \|\mathbf{b}\|\mathbf{a}+\|\mathbf{a}\|\mathbf{b}$

since the sum of two vectors is equal to the diagonal of the parallogram spanned by the two vectors. The two vectors $$\|\mathbf{b}\|\mathbf{a}$$ and $$\|\mathbf{a}\|\mathbf{b}$$ have the same length and therefore span a rhombus. The diagonal of a rhombus cuts the angle exactly in two halves. Alternatively, we could also normalize the two vectors, so both have the length 1 and also span a rhombus.

$\mathbf{c}= \hat{\mathbf{a}}+\hat{\mathbf{b}}$

Each multiple $$k\mathbf{c}$$ has the same property and by setting $$k^{-1}=\|\mathbf{a}\|\|\mathbf{b}\|$$ the special case $$\mathbf{c} = \|\mathbf{b}\|\mathbf{a}+\|\mathbf{a}\|\mathbf{b}$$ from the beginning follows. $$\mathbf{c}$$ is not a unit vector and must be normalized for a proper use.

« Back to Book Overview