REFERENCE : My answer here How to add together non-parallel rapidities?.
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From above reference consider the velocity vectors
\begin{align}
\mathbf{u}_1 \boldsymbol{=} \left(u_{1x},u_{1y},u_{1z}\right) & \boldsymbol{=}\left(u_1 n_{1x},u_1 n_{1y},u_1 n_{1z}\right) \boldsymbol{=} u_1 \mathbf{n}_1\,, \:\: u_1 \in \left(-c,0\right)\cup\left(0,c\right)
\tag{01a}\label{01a}\\
\Vert \mathbf{n}_1 \Vert^2 & \boldsymbol{=} n^2_{1x}\boldsymbol{+}n^2_{1y} \boldsymbol{+} n^2_{1z} \boldsymbol{=}1
\tag{01b}\label{01b}\\
\gamma_1 & \boldsymbol{=} \left(1\boldsymbol{-}\dfrac{u^1_1}{c^2}\right)^{\boldsymbol{-}\frac12}\boldsymbol{=}\cosh\zeta_1
\tag{01c}\label{01c}
\end{align}
and
\begin{align}
\mathbf{u}_2 \boldsymbol{=}\left(u_{2x'},u_{2y'},u_{2z'}\right) & \boldsymbol{=} \left(u_2 n_{2x'},u_2 n_{2y'},u_2 n_{1z'}\right) \boldsymbol{=} u_2 \mathbf{n}_2\,, \:\: u_2 \in \left(-c,0\right)\cup\left(0,c\right)
\tag{02a}\label{02a}\\
\Vert \mathbf{n}_2 \Vert^2 & = n^2_{2x'}+n^2_{2y'} + n^2_{2z'} = 1
\tag{02b}\label{02b}\\
\gamma_2 & \boldsymbol{=} \left(1\boldsymbol{-}\dfrac{u^2_2}{c^2}\right)^{\boldsymbol{-}\frac12}\boldsymbol{=}\cosh\zeta_2
\tag{02c}\label{02c}
\end{align}
Note that $u_1,u_2$ are not the positive magnitudes of $\:\mathbf{u}_1,\mathbf{u}_2$. They are real numbers, that is they can have negative values.
The derived equation
\begin{equation}
\mathbf{u} \boldsymbol{=}\dfrac{\mathbf{u}_2\boldsymbol{+}\dfrac{\gamma^2_{1}\left(\mathbf{u}_1\boldsymbol{\cdot}\mathbf{u}_2\right)}{c^2 \left(\gamma_{1}\boldsymbol{+}1\right)}\mathbf{u}_1\boldsymbol{+}\gamma_1 \mathbf{u}_1}{ \gamma_1\left(1\boldsymbol{+}\dfrac{\mathbf{u}_1\boldsymbol{\cdot}\mathbf{u}_2}{c^{2}}\right)}
\tag{03}\label{03}
\end{equation}
beyond to be the transformation law for 3-velocities, is the law of relativistic addition of 3-velocities, more exactly it's the relativistic sum of $\:\mathbf{u}_1,\mathbf{u}_2$.
For the $\gamma-$factors we have
\begin{equation}
\gamma \boldsymbol{=}\gamma_{1}\gamma_{2}\left(1\boldsymbol{+}\dfrac{\mathbf{u}_1\boldsymbol{\cdot}\mathbf{u}_2}{c^2}\right)
\tag{04}\label{04}
\end{equation}
which from the definition of rapidities
\begin{equation}
\tanh\zeta_1\stackrel{\texttt{def}}{\boldsymbol{=}}\dfrac{u_1}{c}\,,\quad \tanh\zeta_2\stackrel{\texttt{def}}{\boldsymbol{=}}\dfrac{u_2}{c}\,,\quad\tanh\zeta\stackrel{\texttt{def}}{\boldsymbol{=}}\dfrac{u}{c}
\tag{05}\label{05}
\end{equation}
yields
\begin{equation}
\cosh\zeta\boldsymbol{=}\cosh\zeta_1\cosh\zeta_2\boldsymbol{+}\underbrace{\left(\mathbf{n}_1\boldsymbol{\cdot}\mathbf{n}_2\right)}_{\cos\omega}\sinh\zeta_1\sinh\zeta_2
\tag{06}\label{06}
\end{equation}
From the definition of rapidity 3-vectors we have
\begin{align}
\mathbf{w}_1 \stackrel{\texttt{def}}{\boldsymbol{=}}\zeta_1\mathbf{n}_1\boldsymbol{=}\zeta_1\dfrac{\mathbf{u}_1}{u_1}\boldsymbol{=}\dfrac{\zeta_1}{c\tanh\zeta_1}\mathbf{u}_1 \quad \boldsymbol{\Longrightarrow}\quad \mathbf{u}_1 & \boldsymbol{=}\dfrac{c\tanh\zeta_1}{\zeta_1}\mathbf{w}_1
\tag{07a}\label{07a}\\
\mathbf{w}_2 \stackrel{\texttt{def}}{\boldsymbol{=}}\zeta_2\mathbf{n}_2\boldsymbol{=}\zeta_2\dfrac{\mathbf{u}_2}{u_2}\boldsymbol{=}\dfrac{\zeta_2}{c\tanh\zeta_2}\mathbf{u}_2 \quad \boldsymbol{\Longrightarrow}\quad \mathbf{u}_2 &\boldsymbol{=}\dfrac{c\tanh\zeta_2}{\zeta_2}\mathbf{w}_2
\tag{07b}\label{07b}\\
\mathbf{w} \stackrel{\texttt{def}}{\boldsymbol{=}}\zeta\mathbf{n}\boldsymbol{=}\zeta\dfrac{\mathbf{u}\hphantom{_2}}{u}\boldsymbol{=}\dfrac{\zeta}{c\tanh\zeta}\mathbf{u} \quad \boldsymbol{\Longrightarrow}\quad \mathbf{u} &\boldsymbol{=}\dfrac{c\tanh\zeta}{\zeta}\mathbf{w}
\tag{07c}\label{07c}
\end{align}
Multiplying by $\gamma_2$ the nominator and denominator of the rhs in equation \eqref{03} and making use of \eqref{04}
\begin{equation}
\mathbf{u}\boldsymbol{=}\dfrac{\gamma_2\mathbf{u}_2\boldsymbol{+}\dfrac{\gamma_2\gamma^2_{1}\left(\mathbf{u}_1\boldsymbol{\cdot}\mathbf{u}_2\right)}{c^2 \left(\gamma_{1}\boldsymbol{+}1\right)}\mathbf{u}_1\boldsymbol{+}\gamma_2\gamma_1 \mathbf{u}_1}{\cosh \zeta}
\tag{08}\label{08}
\end{equation}
From equations \eqref{07a},\eqref{07b}
\begin{align}
\gamma_1 \mathbf{u}_1 & \boldsymbol{=}\cosh\zeta_1 \mathbf{u}_1\boldsymbol{=}\dfrac{c\sinh\zeta_1}{\zeta_1}\mathbf{w}_1
\tag{09a}\label{09a}\\
\gamma_2 \mathbf{u}_2 & \boldsymbol{=}\cosh\zeta_2 \mathbf{u}_2\boldsymbol{=}\dfrac{c\sinh\zeta_2}{\zeta_2}\mathbf{w}_2
\tag{09b}\label{09b}
\end{align}
Inserting above expressions of $\gamma_1 \mathbf{u}_1,\gamma_2 \mathbf{u}_2$ in equation \eqref{08} and making use of \eqref{07c} we have in terms of rapitidies and rapitidies 3-vectors
\begin{equation}
\dfrac{c\sinh\zeta}{\zeta}\mathbf{w} \boldsymbol{=}\dfrac{c\sinh\zeta_2}{\zeta_2}\mathbf{w}_2 \boldsymbol{+}\dfrac{c\sinh^2\zeta_1\sinh\zeta_2}{\zeta^2_1\zeta_2 \left(\cosh\zeta_1\boldsymbol{+}1\right)}\left(\mathbf{w}_1\boldsymbol{\cdot}\mathbf{w}_2\right)\mathbf{w}_1\boldsymbol{+}\cosh\zeta_2\dfrac{c\sinh\zeta_1}{\zeta_1}\mathbf{w}_1
\tag{10}\label{10}
\end{equation}
and in terms of rapitidies and unit 3-vectors
\begin{equation}
(\sinh\zeta)\mathbf{n} \boldsymbol{=}(\sinh\zeta_1\cosh\zeta_2)\mathbf{n}_1 \boldsymbol{+}(\sinh\zeta_2)\mathbf{n}_2 \boldsymbol{+}\dfrac{\sinh^2\zeta_1\sinh\zeta_2}{ \left(\cosh\zeta_1\boldsymbol{+}1\right)}\left(\mathbf{n}_1\boldsymbol{\cdot}\mathbf{n}_2\right)\mathbf{n}_1
\tag{11}\label{11}
\end{equation}
or
\begin{equation}
(\sinh\zeta)\mathbf{n} \boldsymbol{=}\left[\dfrac{\overbrace{\left[\!\![\cosh\zeta_1\cosh\zeta_2\boldsymbol{+}\left(\mathbf{n}_1\boldsymbol{\cdot}\mathbf{n}_2\right)\sinh\zeta_1\sinh\zeta_2]\!\!\right]}^{\cosh\zeta}\boldsymbol{+}\cosh\zeta_2}{ \left(\cosh\zeta_1\boldsymbol{+}1\right)}\right](\sinh\zeta_1)\mathbf{n}_1 \boldsymbol{+}(\sinh\zeta_2)\mathbf{n}_2
\tag{12}\label{12}
\end{equation}
