# Norm quadrics and their motives

Let $$k$$ be a field of characteristic $$\neq 2$$ and $$\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$$ a Pfister form over $$k$$. Denote by $$Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$$ the projective quadric of dimension $$2^{n-1}-1$$ given by the equation $$\langle\!\langle a_{1},\cdots,a_{n-1}\rangle\!\rangle=a_{n}t^{2}$$. Let $$P_{\underline{a}}$$ denote the quadric given by the equation $$\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle=0$$.

Question: If $$P_{\underline{a}}$$ has a $$k$$-rational point then does $$Q_{\underline{a}}$$ have a $$k$$-rational point?

This question is Lemma 4.2 in “Motivic cohomology with $$\mathbb{Z}/2$$-coefficients” by V. Voevodsky. I do not understand this proof; for any rational point $$p$$ of $$P_{\underline{a}}$$, why can he say that there exists a linear subspace $$H$$ of dimension $$2^{n-1}-1$$ which lies on $$P_{\underline{a}}$$ and passes through $$p$$?