A satellite orbits a planet. At point $A$, its speed is $v_1$​. At point $B$, its speed is $v_2$​ and its velocity vector forms a right angle with the velocity vector at point $A$. At point $C$ the velocity is exactly opposite to the velocity at point $A$, with a magnitude of $v_3$​.

Find the eccentricity of the orbit.

Also determine the exact numerical value of the eccentricity when $v_1​=1~\text{km}/\text{s}$, $v_2=2~\text{km}/\text{s}$, $v_3=3~\text{km}/\text{s}$.