You can do rotations and translations "via linear algebra" without homogeneous coordinates (the 4-element tuple representing a 3d point or vector), since adding two vectors is linear algebra.

What this lets you do is composing multiple transforms into one matrix multiplication instead of a sequence of multiplications and additions; that's what dramatically increases performance, on modern computing devices but most especially on ancient ones, where we were fighting for every MUL.

More details: https://gabrielgambetta.com/computer-graphics-from-scratch/1...