Given $S^+ = \{(x,y,z) \in \mathbb{R}^3: -x^2 - y^2 + z^2 =1, z > 0\},$ if (1) the hyperboloid norm of $v$ is defined to be $$||v||_{S^+,(x,y,z)} = \sqrt{a^2 + b^2 - c^2,}$$ where $(a, b, c)$ is the vector tangent to the point $(x,y,z)$ in $S^+,$ (2) the length of a differentiable curve $t \rightarrow \gamma(t), t_0 \leq t \leq t_1$ in $S^+$ is defined $$l_{S^+}(\gamma) = \int_{t_0}^{t_1} ||\gamma'(t)||_{S^+,\gamma(y)}dt,$$ and the distance $d_{S^+}(p,q)$ between two points $p$ and $q$ in $S^+$ is defined $$d_{S^+}(p,q) = \inf\{l_{S^+}(\gamma): \gamma \textrm{ is a piecewise differentiable curve in $S^+$ from $p$ to $q$}\}$$
how does it follow that $$\cosh(d_{S^+}(p,q)) = -x_1x_2 - y_1y_2 + z_1z_2,$$ where $p = (x_1,y_1,z_1)$ and $q = (x_2, y_2, z_2).$
Can I get a hint? Thanks!