Minkowski-space/Two observers/Relative velocity (vector)/Definition

Let ${\displaystyle {}V}$ denote a Minkowski space, and let ${\displaystyle {}B}$ and ${\displaystyle {}C}$ denote observers with the four-velocities ${\displaystyle {}v}$ and ${\displaystyle {}w}$. The vector

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

is called the velocity vector of ${\displaystyle {}C}$ relative to ${\displaystyle {}B}$. The number

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

is called the relative velocity of the two observers.