Given a vector \(\mathbf{x}\) in \(\RNrSpc{n}\), and a number \(i\), \(1 \leq i \leq n\), the \(i\)-th direction angle of \(\mathbf{x}\) is
\(\omega_i\ \DefEq\ \sphericalangle(\mathbf{e}_i,\mathbf{x})\)
Here we develop the basic facts about the angles of a vector \(\mathbf{x}\) in \(\RNrSpc{n}\) and any of the coordinate axes.
Given a vector \(\mathbf{x}\) in \(\RNrSpc{n}\), and a number \(i\), \(1 \leq i \leq n\), the \(i\)-th direction angle of \(\mathbf{x}\) is
\(\omega_i\ \DefEq\ \sphericalangle(\mathbf{e}_i,\mathbf{x})\)
To compute direction angles we use the dot product, as in proposition below.
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 cosines of the direction angles of a given vector are related by the following formula.
The sum of cosines of direction angles \(\omega_i\) of a nonzero vector \(\mathbf{x}\) in \(\RNrSpc{n}\) satisfy
\[\cos^2 \omega_1\ +\ \cos^2 \omega_2\ +\ \cdots\ +\ \cos^2 \omega_n\ =\ 1\]