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 \(k=\|\mathbf{a}\|\|\mathbf{b}\|\) 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.