  The open unit ball $\mathbb{B} =
\{\mathbf{v}\in\mathbb{R}^n\colon\|\mathbf{v}\|<1\}$ is endowed with M\"{o}bius
addition $\oplus_M$ defined by $$\mathbf{u}\oplus_M\mathbf{v} = \dfrac{(1 +
2\langle\mathbf{u},\mathbf{v}\rangle + \|\mathbf{v}\|^2)\mathbf{u} + (1 -
\|\mathbf{u}^2)\mathbf{v}}{1 + \langle\mathbf{u},\mathbf{v}\rangle +
\|\mathbf{u}\|^2\|\mathbf{v}\|^2\|}$$ for all $\mathbf{u},\mathbf{v}\in
\mathbf{B}$. In this article, we prove the inequality $$
\dfrac{\|\mathbf{u}\|-\|\mathbf{v}\|}{1+\|\mathbf{u}\|\|\mathbf{v}\|}\leq
\|\mathbf{u}\oplus_M \mathbf{v}\| \leq
\dfrac{\|\mathbf{u}\|+\|\mathbf{v}\|}{1-\|\mathbf{u}\|\|\mathbf{v}\|} $$ in
$\mathbb{B}$. This leads to a new metric on $\mathbb{B}$ defined by
$$d_T(\mathbf{u},\mathbf{v}) = \tan^{-1}{\|-\mathbf{u}\oplus_M\mathbf{v}\|},$$
which turns out to be an invariant of M\"{o}bius transformations on
$\mathbb{R}^n$ carrying $\mathbb{B}$ onto itself. We also compute the isometry
group of $(\mathbb{B}, d_T)$ and give a parametrization of the isometry group
by vectors and rotations.
