Two 3D vectors of same magnitude are given. Only their unit vector can me manipulated. What would be the average value of all possible relative velocity vectors(Difference of the two vectors) formed by the two vectors of same magnitude(take magintude of initail vectors as x)?
The following question was asked my my professor with no more context
I tried taking two fixed vectors first but could not comprehend result for all directions
My professor also told to think what could be the average unit vector of our relative velocity like vector.
Let $\vec{u}$ denote the fixed vector and $\vec{v}$ denote the varying vector. We are interested in the average of the quantity $\vec{w} = \vec{v}-\vec{u}$ where $\vec{v}$ is uniformly distributed over a sphere.
Notice that since $\vec{v}$ is uniformly distributed over a sphere, its average must be the zero vector by symmetry. The average of the fixed vector $\vec{u}$ is just itself. Taking averages (expectation) is a linear operation, so we simply add these averages to obtain
$$\vec{w}_{av} = \vec{v}_{av} - \vec{u}_{av} = \vec{0} - \vec{u} = -\vec{u}.$$