The \(i\)-th direction angle of a vector \(\mathbf{x} = (x_1,\dots ,x_n)\) in \(\RNrSpc{n}\) is
\[\omega_i\ =\ \arccos \frac{x_i}{| \mathbf{x} |}\]The \(i\)-th direction angle of a vector \(\mathbf{x} = (x_1,\dots ,x_n)\) in \(\RNrSpc{n}\) is
\[\omega_i\ =\ \arccos \frac{x_i}{| \mathbf{x} |}\]To see where this identity comes from, recall that \(\mathbf{e}_i\) has length \(1\); i.e. \(| \mathbf{e}_i | = 1\). Then we compute
\[x_i = \mathbf{e}_i \bullet \mathbf{x} = | \mathbf{e}_i | | \mathbf{x} | \cos \omega_i = | \mathbf{x} | \cos \omega_i\]This implies the claim.