Sure, encoding an extra dimension in a vector is just an additional element, but for the exception of categorical data this view is very restrictive. If you want to do things like describe embedded-space and projective spaces you can't just add a term to your formulas and expect everything to work. Like an ant walking on a ball in your room on earth in spacetime projected on your computer screen.

In geometric algebra there is a way to encode every element and transformation in such space and those correspond to shuffling around terms in an equation.