To add to the other responses at this level, I want to point out that one form of vector-vector "multiplication"—inner products—corresponds to applying linear functionals, i.e., to linearly mapping a vector space into its underlying scalar field.
So just as every matrix is a representation of a linear transformation of vectors into other vectors, with matrix-vector multiplication corresponding to function application, it is also true that each vector in a vector space represents a linear transformation that turns vectors in the space into a scalar, with vector-vector multiplication in the form of inner products corresponding to function application. The converse is also true: every linear functional on a vector space can be represented by a vector in the space.
This last insight is known (in various forms) as the Riesz representation theorem and holds not only on finite inner-product spaces (i.e., vector spaces on which an inner product is defined) but also on Hilbert spaces (complete inner product spaces, whether finite or infinite). It turns out to be quite powerful.
So just as every matrix is a representation of a linear transformation of vectors into other vectors, with matrix-vector multiplication corresponding to function application, it is also true that each vector in a vector space represents a linear transformation that turns vectors in the space into a scalar, with vector-vector multiplication in the form of inner products corresponding to function application. The converse is also true: every linear functional on a vector space can be represented by a vector in the space.
This last insight is known (in various forms) as the Riesz representation theorem and holds not only on finite inner-product spaces (i.e., vector spaces on which an inner product is defined) but also on Hilbert spaces (complete inner product spaces, whether finite or infinite). It turns out to be quite powerful.