Why is vector times vector equal to a number? It just occurred to me that we have
$$
\text{number} \cdot \text{number} = \text{number} \\
\text{matrix} \cdot \text{matrix} = \text{matrix}
$$
but
$$
\text{vector} \cdot \text{vector} = \text{number}
$$
Why is that? Why is $\text{vector} \cdot \text{vector}$ not equal to another $\text{vector}$? Is that just a historical accident, that the sign "$\cdot$" is used that way for vectors, or is there a deeper reason for this difference in multiplication between numbers, matrices and vectors?

Note that the product vector by vector is a special case of the product matrix by vector indeed given a $m$ by $n$ matrix $A$ and a $n$ by $1$ vector $v$ the product is a $m$ by $1$ vector $w$ and thus the product of a $1$ by $n$ vector $u$ by a $n$ by $1$ vector $v$ is a $1$ by $1$ matrix that is a number. Such kind of product is also known as dot product
$$u\cdot v= u^Tv=\sum u_iv_i$$
As noticed, in three dimensions we can also define the cross product by vectors which returns another vector.
