Finding a bisector hyperplane of two vector in n-dimensional space

Question

Suppose two vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ in n-dimensional space. I want to find a hyperplane that bisects the angle between these two vectors.

Answer 1

Normalize (really, they just need to be the same length): $$ \hat v_1 = \frac{v_1}{|v_1|},\quad \hat v_2 = \frac{v_2}{|v_2|}. $$ Then $w = \hat v_2 - \hat v_1$ is normal to the hyperplane $H$ you want: $$ H = \{x \in \mathbb R^n \;:\; x\cdot w = 0\}, $$ where $\cdot$ is the standard inner product on $\mathbb R^n$.

Answer 2

I am not sure what do you mean by a bisector hyperplane. But assuming the standard metric of $\mathbb R^n$ I think that you are looking for the following construction:

• Normalize $v_1$ and $v_2$, defining $u_1:=\frac{v_1}{\parallel v_1 \parallel}$ and $u_2:=\frac{v_2}{\parallel v_2 \parallel}$
• Consider a basis $\{b_1,\ldots,b_{n-2}\}$ of the orthogonal complement to $\langle v_1, v_2\rangle$.
• The required hyperplande would be $$ \langle b_1,\ldots,b_{n-2},u_1+u_2 \rangle $$