Methane In \(\mathbb{R}^4\)?

Geometry Level 5

Consider five nonzero vectors in \(\mathbb{R}^4\) such that any two of them enclose the same nonzero angle \(\theta\). Find \(\cos\theta\), to two significant digits.

Bonus Question: Find the corresponding angle \(\theta_n\) between \(n+1\) nonzero vectors in \(\mathbb{R^n}\) and find \(\displaystyle \lim_{n\to\infty}\theta_n\).

Note that \(\theta_3\) is the bond angle in the methane molecule.

Image Credit: C.A.G.E

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