Solution
= Solution
Let $B$ be the rank-two tensor and $v$ the vector. The components of their <tensor product> are $T_{ijk}=B_{ij}v_k$. Applying the transformation rules gives
$$
T'_{ijk}=B'_{ij}v'_k=Q_{ia}Q_{jb}Q_{kc}B_{ab}v_c=Q_{ia}Q_{jb}Q_{kc}T_{abc}.
$$
The product therefore obeys the three-index <tensor component transformation law>, proving that \b[$B_{ij}v_k$ is a rank-three tensor]. Tensor-product order adds because no pair of indices is summed out.