I'm quoting what I found in a book about quantum computation:
Individual state spaces of $n$ particles combine quantum mechanically through the tensor product. If $X$ and $Y$ are vectors, then their tensor product $X\otimes Y$ is also a vector, but its dimension is $\dim(X) \times \dim(Y)$, while the vector product $X\times Y$ has dimension $\dim(X)+\dim(Y)$. For example, if $\dim(X)= \dim(Y)=10$, then the tensor product of the two vectors has dimension $100$, while the vector product has dimension $20$.
I don't understand: how can he state that the result of a vector product has dimension $\dim(X) + \dim(Y)$? What does he intend for dim?