I was reading Apostol and in it's $12^{th}$ chapter (page 446) it said-
Unfortunately, the geometric pictures which are a great help in motivating and illustrating vector concepts when $n = 1, 2,$ and $3$ are not available when $n > 3$; therefore, the study of vector algebra in higher-dimensional spaces must proceed entirely by analytic means.
So, I was wondering what would have been the state of mathematics if we had geometric pictures of vector spaces with $>3$ dimensions? Would we be able to solve more geometric problems? I think we still would have the same amount of results but enriched by their geometric understanding. Right?
We get so much insight from 1D/2D/3D graphical diagrams (example) that there is no much doubt on my side that a 4D canvas would give additional insight if we were beings of 4 spatial dimensions with appropriate senses and intellect. This of course includes 4D vector algebra.