Do $4$ orthogonal lines exist in $\mathbb{R}^4$?

Question

I just learned about the cross product in linear algebra.

I need some help with a mental image.

In math, obviously not in our $\mathbb{R}^3$ world, do there exist $4$ orthogonal lines in $\mathbb{R^4}$? And how could one visualize that in $\mathbb{R^3}$? Is it possible?

My intuition says it should be ...

Br, Twoface

Answer 1

Think of the coordinate axes as the 4 lines. Since we live in 3 dimensions, the only way to visualize it is by looking at "snapshots," or contours. Draw 4 different 3d coordinated axes, and just permute which axes you're looking at. For example, have one set the x,y,z axes (this is equivalent to setting w = C) and another the x,y,w axes (setting z = C). This may be easier to do if you had a function and were able to see what each contour looks like.