Solution
= Solution
Contract with $v_j$ and use its squared <Euclidean norm>:
$$
\boxed{P_{ij}v_j=v_i-v_i\frac{v_jv_j}{v_kv_k}=0.}
$$
Thus $v$ lies in the <kernel> of this <orthogonal projection>.
= Solution
Contract with $v_j$ and use its squared <Euclidean norm>:
$$
\boxed{P_{ij}v_j=v_i-v_i\frac{v_jv_j}{v_kv_k}=0.}
$$
Thus $v$ lies in the <kernel> of this <orthogonal projection>.