Is is possible for a hyperplane with a normal vector in $\mathbb{R^n} \subset \mathbb{R^{n+1}}$ to intersect a point with some nonzero $x_{n+1}$ coordinate?
The resulting equation for the plane does not even contain a $n+1$ coordinate which makes it impossible in my opinion.
My text is talking about a vertical hyperplane in $\mathbb{R^{n+1}}$ being generated by a vector in $\mathbb{R^n}$ but I find that hard to imagine.
Who's right?
To imagine it, take $n=2$, and consider a vertical hyperplane $H$ in $\mathbb{R}^3$.
The projection of $H$ onto the $xy$-plane (regarded as $\mathbb{R}^2$) is a line in the $xy$-plane (which is a hyperplane relative to $\mathbb{R}^2$).
Does that help with the visualization?