Is there a geometric meaning to the outer product of two vectors?

Question

Define two vectors v and u in $\mathbb{R}^3$. I know the geometric meaning of the inner and cross product.

Is there a meaning to the matrix resulting from $\textbf{uv}^T$?

Answer 1

For any vector $x$, $$ uv^T(x) = (v \cdot x)u $$ That is, If $u$ and $v$ are unit vectors, $uv^T(x)$ is the component of $x$ in the $v$ direction, taken into the $u$ direction.

This interpretation makes for a neat understanding of singular value decomposition.