[QUE/ME-02002] ME-PROBLEM

  The coordinates of a point under rotation change as \begin{equation} \label{EQ01} \begin{pmatrix} x_1{'}\\ x_2{'}\\ x_3{'}\end{pmatrix} = R \begin{pmatrix} x_1\\ x_2\\ x_3\end{pmatrix} \end{equation} Show that the property that lengths and angles do not change under a rotation implies that \(R\) must be an orthogonal matrix. Hence prove that \(\det R\) can have only value equal to \(\pm1\). What condition(s) on the transformation \eqRef{EQ01} will lead to the requirement that \(\det R\) must be equal to 1 and that \(\det R= -1\) will be ruled out?