Minkowski-space/Two observers/Relative velocity/Fact/Proof

${\displaystyle {}\left\langle v,v_{BC}\right\rangle =\left\langle v,-v-{\frac {1}{\left\langle v,w\right\rangle }}w\right\rangle =\left\langle v,-v\right\rangle +\left\langle v,-{\frac {1}{\left\langle v,w\right\rangle }}w\right\rangle =1-1=0\,,}$

so these vectors are orthogonal to each other. Therefore, ${\displaystyle {}v_{BC}}$ belongs to the space component of ${\displaystyle {}B}$.

${\displaystyle {}{\begin{aligned}\left\langle v_{BC},v_{BC}\right\rangle &=\left\langle -v-{\frac {1}{\left\langle v,w\right\rangle }}w,-v-{\frac {1}{\left\langle v,w\right\rangle }}w\right\rangle \\&=\left\langle v+{\frac {1}{\left\langle v,w\right\rangle }}w,v+{\frac {1}{\left\langle v,w\right\rangle }}w\right\rangle \\&=\left\langle v,v\right\rangle +2{\frac {\left\langle v,w\right\rangle }{\left\langle v,w\right\rangle }}+{\frac {1}{\left\langle v,w\right\rangle ^{2}}}\left\langle w,w\right\rangle \\&=1-{\frac {1}{\left\langle v,w\right\rangle ^{2}}}.\end{aligned}}}$

By part (1) (or because of fact  (3)), this expression is non-negative. The square root of this number is the relative velocity ${\displaystyle {}\rho }$.

${\displaystyle {}\rho ={\sqrt {1-{\frac {1}{\left\langle v,w\right\rangle ^{2}}}}}\,,}$

using

${\displaystyle {}\left\langle v,w\right\rangle <0\,.}$

${\displaystyle {}v_{BC}=-v-{\frac {1}{\left\langle v,w\right\rangle }}w\,}$

and (3), we can infer

${\displaystyle {}w=-\left\langle v,w\right\rangle v-\left\langle v,w\right\rangle v_{BC}={\frac {v}{\sqrt {1-\rho ^{2}}}}+{\frac {v_{BC}}{\sqrt {1-\rho ^{2}}}}\,.}$

Because of part (1), the vector ${\displaystyle {}v_{BC}}$ belongs to the space component of ${\displaystyle {}B}$.