I am working on a problem which requires me to find certain values of the components of vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^4$ such that the angle between them is $\pi/3$ If my understanding is correct then the answer should be the values for which $\arccos{((u\cdot v)/(|u| \cdot |v|))} = \pi/3$
But I am having a little trouble understanding what an angle means in higher dimensional spaces such as $\mathbb{R}^4$. So what is the definition of an angle in $n$ dimensions? Am I misunderstanding the problem?