Specification of 2D plane in 4D space

Question

My understanding is that for a given point of interception on a line in 3D space there is one plane such that all of its points are orthogonal to that line. Is there the same 1:1 correspondence in 4D space, or is there more than one set of points orthogonal to a given line at a given point of interception on it?

Answer 1

There’s still only one set of points orthogonal to a given line at each point on the line, but that set is now three-dimensional: it’s a hyperplane. If you want to specify a two-dimensional plane in 4D using lines perpendicular to it, you’ll need another line that’s not parallel to the first to select from all of the two-dimensional planes that are orthogonal to the first line. Essentially, you’d be describing this plane as the intersection of two hyperplanes.

Answer 2

Given a line in a four-dimensional Euclidean space and a point on that line, there is a three-dimensional hyperplane perpendicular to that line passing through that point. The hyperplane contains many two-dimensional planes, all of which also are perpendicular to the line.

So there is a one-to-one relationship in four-dimensional space like the one you described in three-dimensional space, but it involves three-dimensional hyperplanes rather than two-dimensional planes.