Note: this section is better understood with an understanding of matrices.
Another fundamental type of vector product is called an outer product. Where the inner product can be represented by the vector product \( y^Tx \in \mathbb{R} \) the outer product is given by switching the order of the row and column vector. $$ xy^T \in \mathbb{R}^{n \times n} $$ An outer product is fundamentally a matrix, consisting of all possible pairwise products of elements of \(x\) and \(y\). Explicitly, we have $$ \Big[ \ xy^T \Big]_{ij} = x_i y_j $$ Note also that we can think of each column of \(xy^T\) as a copy of \(x\) each scaled by an element of \(y\) For two vector \(x\) and \(y\) (in two, three, and four dimensions) the columns of \(xy^T\) are shown below. Note that each column lies in the same 1D subspace defined by \(x\). Note also that columns are a parallel axis representation of vector \(y\) (in the \(x\) direction and scaled by the length of \(x\)). The rows would have a similar geometry with the roles of \(x\) and \(y\) reversed. The outer product is a rank-1 matrix or a dyad. It has a 1D range and a nullspace with dimension \( n-1 \).