Question

Find the angle between a diagonal of a 4-Dimensional hypercube and a diagonal of one of it's faces.

Answer 1

The diagonal of a unit 4D hypercube with a corner at the origin goes from the point \((0,0,0,0)\) to the point \((1,1,1,1)\), so we can represent this as a vector \(a = \langle 1,1,1,1\rangle\). A diagonal on one of its faces (which I'm assuming means a volume) could be chosen to go from the point \((0,0,0,0)\) to \((0,1,1,1)\) so we can represent this also as a vector \(b = \langle 0,1,1,1\rangle\).

Using the law of cosines, we have:

\[a \cdot b = |a||b| \cos \theta\] so the angle we want is:

\[\cos^{-1} \left( \frac{a \cdot b}{|a||b|} \right) =\cos^{-1} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{6} \]

Answer 2

Thanks a lot!