I hope this is the right forum... In 3d space, a 2d plane can be specified by it's normal vector. In 4d space, is there a 3d plane, and will these planes be specifiable by a single vector?
No, that's not enough information. You can specify a plane in R^{3} by its normal vector and a point on the plane. Without that point what you get is a family of parallel planes. In higher dimensions, including R^{4}, we call them hyperplanes. And again, a single vector isn't enough.
In general, we can specify a n-1 dimensional hyperplane in a space of n dimensions with a "normal vector" and a point in the hyperplane. In four dimensions, every point can be written as [itex](x_1, x_2, x_3, x_4)[/itex] and a four dimensional vector of the form [itex]a\vec{ix}+ b\vec{j}+ c\vec{k}+ d\vec{l}[/itex]. If the origin, (0, 0, 0, 0) is in the hyperplane, then we can write [itex]x_1\vec{ix}+ x_2\vec{j}+ x_3\vec{k}+ x_4\vec{l}[/itex] and so the dot product is [tex]ax_1+ bx_2+ cx_3+ dx_4= 0[/tex] giving an equation for that hyper plane. But, again, that assumes the hyperplane contains the point (0, 0, 0). Another plane, perpendicular to the same vector, but not containing (0, 0, 0), cannot be written that way.