3d – New coordinate of another point after rotating one point on $S^2$?

As we rotate $p_a = (x_a, y_a, z_a)$ to $p_b = (x_b, y_b, z_b)$ (both are known), how can we find the the new location of $p_c = (x_c, y_c, z_c)$ after this rotation? Can we instead find out the new location of $overrightarrow{p_a p_c}$ after this rotation? Thanks in advance.