What mathematical relationship connects the total space velocity (V) using the radial velocity ($V_r$) and tangential velocity ($V_t$)?

The Pythagorean theorem: $V^2 = V_r^2 + V_t^2$

The total space velocity ($V$) represents the magnitude of the star's motion in three dimensions, which is calculated by combining its two perpendicular components: the radial velocity ($V_r$) and the tangential velocity ($V_t$). Since the radial velocity is defined as movement directly along the line of sight, and the tangential velocity is defined as movement strictly perpendicular to that line of sight, these two vectors form the legs of a right triangle when visualized. Therefore, the hypotenuse, which represents the total space velocity, must be calculated using the Pythagorean theorem, resulting in the relationship $V^2 = V_r^2 + V_t^2$. This geometric combination allows astronomers to find the actual speed of the star relative to the observer once both directional components have been independently measured or derived.