Multiplication of sum of vectors

Question

With x and y are vectors, can I do normal algebra to them like this: (x+y)(x+y) = xx + 2(xy) + yy Is that valid?

Answer 1

Not generally, no. You need to define a multiplication of vectors, and moreover from your formula you'd need that multiplication to be commutative.

For instance, this works on $\mathbb{R}$ or $\mathbb{C}$, but not on $\mathbb{R}^2$ or a space of matrices.

Answer 2

It depends on what product you use.

With vectors you have a couple of useful product operations, which resemble but are not identical to the product you are used to from fields like real or complex numbers.

Examples are scalar product, vector product, scalar multiplication