Let ##P:=\mathbb{R}\cdot \vec{p} \oplus \mathbb{R}\cdot \vec{q}## be the plane and ##\vec{n}## its normal vector. Then ##\langle \vec{n},P \rangle =0## by definition of normality. This means that ##\langle \vec{n},\lambda \vec{p}+\mu \vec{q} \rangle=0## for all ##\lambda ,\mu \in \mathbb{R}.## This is especially true for the given vector contained in the plane whose coordinates are a specific pair ##(\lambda ,\mu).##
If you define normality by ##\vec{n} \perp \vec{p}\, \wedge \vec{n}\perp \vec{q} \,,## then we get
$$
0=\lambda \cdot 0+\mu\cdot 0=\lambda \langle \vec{n},\vec{p}\rangle +\mu \langle \vec{n},\vec{q}\rangle=\langle \vec{n},\lambda \vec{p}+\mu\vec{q}\rangle
$$
and again ##\vec{n}\perp \lambda \vec{p}+\mu\vec{q}## for a given pair of coordinates ##(\lambda ,\mu).##