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?

Three kinds of vector products, along with what they produce:


*

*Dot product: $vector \cdot vector = scalar$

*Cross product: $vector \times vector = vector$

*Outer product: $vector \otimes vector = matrix$


So, it only produces a number (scalar) if it's a dot product.
It boils down to definitions.
