Given three (non-zero) linearly independent vectors $u, v, w$ in $\mathbb{R}^3$ and form the parallelpiped spanned by these three vectors. Is there a way to compute the dimensions of the smallest rectangular prism that contains this parallelpiped?
In two dimensions, if we're given two vectors $v, w$ in $\mathbb{R}^2$, the parallelogram spanned by these two vectors is contained in a rectangle with length $\|v + w\|$ and height equal $2\|w - \mathrm{proj}_{v + w}(w)\|$ where $\mathrm{proj}_{v + w}(w)$ is the projection of $w$ onto $v + w$.
Is there a similar result in $\mathbb{R}^3$?