Calculating geodesics $\mathbb{R}^3-\{(0,0,0)\}$

Question

I have to calculate the geodesics of the Riemann manifold $(\mathbb{R}^3-\{(0,0,0)\},g)$, where $g$ is the Riemann metric $$ g=\frac{dx^2+dy^2+dz^2}{x^2+y^2+z^2} $$ I'm supposed to use that the diffeomorphism $\psi:S^2\times\mathbb{R}\rightarrow \mathbb{R}^3-\{(0,0,0)\}$, $\psi((v_1,v_2,v_3),t)=e^t(v_1,v_2,v_3)$ is an isometry, when we use the product metric in $ S^2\times\mathbb{R}$ and $g$ in $\mathbb{R}^3-\{(0,0,0)\}$. Is it possible to calculate those geodesics using this isometry with out calculating the Christoffel symbols?