Soit \(\vec V = a \vec x_2 + b \vec y_2 + c \vec z_2\).
Exprimer \(\vec V\) dans la base \(b_1\).
\(\vec V = {\begin{array}{c} \\ \\ \\ \end{array}}_{b_1} \left| \begin{array}{c} a \ \cos \theta + b \ \sin \theta\\ a \ \sin \theta + b \ \cos \theta\\ c\end{array}\right.\)
\(\vec V = {\begin{array}{c} \\ \\ \\ \end{array}}_{b_1} \left| \begin{array}{c} a \ \cos \theta - b \ \sin \theta\\ a \ \sin \theta + b \ \cos \theta\\ c\end{array}\right.\)
\(\vec V = {\begin{array}{c} \\ \\ \\ \end{array}}_{b_1} \left| \begin{array}{c} a \ \cos \theta - b \ \sin \theta\\ - a \ \sin \theta + b \ \cos \theta\\ c\end{array}\right.\)
\(\vec V = {\begin{array}{c} \\ \\ \\ \end{array}}_{b_2} \left| \begin{array}{c} a \ \cos \theta - b \ \sin \theta\\ - a \ \sin \theta + b \ \cos \theta\\ c\end{array}\right.\)
\(\vec V = {\begin{array}{c} \\ \\ \\ \end{array}}_{b_1} \left| \begin{array}{c} - a \ \cos \theta - b \ \sin \theta\\ a \ \sin \theta + b \ \cos \theta\\ c\end{array}\right.\)