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\(\newcommand{\Prime}{^\prime}\)The coordinates of a point under rotation change as \begin{equation} \begin{pmatrix} x_1\Prime\\ x_2\Prime\\ x_3\Prime\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 imples that \(R\) must be an orthogonal matrix. Hence show that \(\det R\) must be \(\pm 1\).
What condition on the transformation will lead to the requirement that \(\det R\) must be equal to 1 and \(\det R=-1\) will ruled out?
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4727:Diamond Point
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