$v^Tv$ is the dot product of a vector with itself, which is just its norm squared, an intuitive geometric quantity.
Is there something to be said about $vv^T$? Is there some kind of relationship between this matrix and some geometric object the way the cross product relates to the area of the parallelogram for example?
Consider an arbitrary vector $\boldsymbol{a}$ and a unit vector $\boldsymbol{v}$.
Then $\boldsymbol{v} \cdot \boldsymbol{a} = \boldsymbol{v}^\intercal \boldsymbol{a}$ is the component of $\boldsymbol{a}$ along $\boldsymbol{v}$.
More specifically the projection of $\boldsymbol{a}$ parallel to $\boldsymbol{v}$
$$ \boldsymbol{a}_\parallel = \boldsymbol{v} \left( \boldsymbol{v}^\intercal \boldsymbol{a} \right)$$
The above can be viewed as a matrix/vector operation
$$ \boldsymbol{a}_\parallel = [ \boldsymbol{v} \boldsymbol{v}^\intercal] \boldsymbol{a} $$
where $[\boldsymbol{v} \boldsymbol{v}^\intercal] = \boldsymbol{v} \odot \boldsymbol{v}$ is the outer product of $\boldsymbol{v}$ with itself and results in a $n\times n$ matrix.
The interpretation is that $[\boldsymbol{v} \boldsymbol{v}^\intercal]$ projects any arbitrary vector $\boldsymbol{a}$ along $\boldsymbol{v}$