The bilinear form $B$ is defined by
$$B:\left(\begin{pmatrix}x\\y\\z\end{pmatrix},\begin{pmatrix}x'\\y'\\z'\end{pmatrix}\right)\mapsto 3xx'+3yy'-zz'.$$
Given a vector $v\in V$, the pseudo-orthogonal of $v$ is $v^\perp = \{w\in V \mid B(v,w)=0\}$.
Let $v$ be a non-zero vector of $V$. Show that $v^\perp$ is a vector subspace of $V$ of codimension 1, and that $v^\perp$ is a complement of the line generated by $v$ if and only if $B(v,v)\neq 0$.