Angle between two vectors in an n-th dimensional space, where n is greater than 3

Question

Suppose A and B are two non-parallel vectors. Where $ A $ and $ B $ are given as $ A =(a_1, a_2, a_3,.....a_n) $ and $ B =(b_1, b_2, b_3,.....b_n) $ How can I geometrically determine the angle $ θ $ between the two vectors A and B? I know that $ A.B = \left\lVert A \right\rVert \left\lVert B \right\rVert \cos(θ) $

Answer 1

\begin{align*} \theta&=\cos^{-1}\left(\frac{A\cdot B}{\left\lVert A \right\rVert \left\lVert B \right\rVert}\right)\\ &=\cos^{-1}\left(\frac{(a_1, a_2, a_3,.....a_n)\cdot (b_1, b_2, b_3,.....b_n)}{\sqrt{\displaystyle\sum_{k=1}^na^2_k}\sqrt{\displaystyle\sum_{k=1}^nb_k^2}}\right)\\& =\cos^{-1}\left(\frac{\displaystyle\sum_{k=1}^na_kb_k}{\sqrt{\displaystyle\sum_{k=1}^na^2_k}\sqrt{\displaystyle\sum_{k=1}^nb_k^2}}\right) \end{align*}