How can you find the maximum height of an object rotated in any direction? I had an 86in long couch I wanted to get into the basement and while the ceiling was 90in, because you couldn't bring it down the stairs keeping it perpendicular to the floor, it had to be rotated and it didn't fit, because once tilted, it was taller than 90in. Sadly I had to donate the couch because I couldn't make it fit.
But, it made me wonder how best to determine what could fit given an object that had to be rotated along an axis. Or rather, if an object were rotated, what would the height be as a function of its rotation.
Say I had an $86 \times 24$ rectangle with a corner touching the origin $(0,0)$ and rotated it $45^\circ$, what would the height of this shape be? What are the steps involved? What if I rotated it $360^\circ$, what is the maximum height that shape might reach throughout the entire transform?
 A: If I understand your situation correctly, the highest point will be obtained with the vector connecting the origin to the farthest point of the shape. For a generic shape, we would choose $v$ in the following way:

The highest height is then simply the length of $v$
For example, if you take a rectangle, this vector would be the one connecting the origin to the diagonal, and after a rotation so that this vector lies on the $y$-axis, we have :

So that the highest height would be given by Patheagoran theorem $h = \sqrt{x^2 + y^2}$
If you want the height as a function of rotation, applying a rotation of angle $\theta$ to $v = (x,y)$, we have
$$ R(\theta) \ v = \begin{pmatrix} \cos \theta &  - \sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \cos \theta - y \sin \theta \\ x \sin \theta + y \cos \theta \end{pmatrix} $$
so that
$$ h (\theta) = x \sin \theta + y \cos \theta $$
It is a good exercise to check that the maximum of $h$ is given by $\sqrt{x^2 + y^2}$, as predicted by our intuitive picture.
