it's directly from the definition.

The null space of A is all vectors $v = \begin{pmatrix}x\\y\\z\end{pmatrix} \ni Av=0$

$A = \begin{pmatrix}1 & 2 &2 \end{pmatrix}$

$\begin{pmatrix}1 &2 &2\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} = 0 \Rightarrow x + 2y + 2z = 0$