I'm trying to understand the Steiner construction generalized to a higher dimensional space. I'm reading Griffiths & Harris. Principles of algebraic geometry, page 529. They propose the following generalization:
If $V_1$, $V_2$ are two $(n-2)$-planes in $\mathbb{P}^n$, we may choose any parametrization of the two pencils of hyperplanes $\{H_1(\lambda)\}$ and $\{H_2(\lambda)\}$ through $V_1$ and $V_2$ respectively such that $H_1(\lambda)\not=H_2(\lambda)$ for all $\lambda$, and consider the locus $$Q=\bigcup\limits_\lambda H_1(\lambda)\cap H_2(\lambda)$$ $Q$ is readily seen to be an irreducible, nondegenerate hypersurface, and hence a quadric, intersecting a general line $L\subset \mathbb{P}^n$ in the fixed points of the automorphism of $L$ sending $L\cap H_1(\lambda)$ to $L\cap H_2(\lambda)$. $Q$ is a quadric of rank either 3 or 4, with vertex $V_1\cap V_2$
First of all, how do I prove that $Q$ is a quadric? Also, why is the rank 3 or 4 and why is $V_1\cap V_2$ the vertex? I would appreciate any help.