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[QUE/ME-02007]

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\(\newcommand{\Prime}{^\prime}\)Let \({\bf A, B,..}\) be objects with components written as \(\vec{A}=(A_1,A_2,A_3), \vec{B}=(B_1,B_2,B_3)\). Introduce \(\vec{A}\Prime=(A_1\Prime,A_2\Prime,A_3\Prime), \vec{B}\Prime=(B_1\Prime,B_2\Prime,B_3\Prime)\) etc. by means of equation \begin{equation}\label{EQ01} \begin{pmatrix}A_1\Prime\\A_2\Prime\\A_3\Prime\end{pmatrix} = R \begin{pmatrix}A_1\\A_2\\A_3\end{pmatrix} \end{equation} and with similar equations for other vectors.

Using the fact that the matrix \(R\) is orthodgonal matrix show that

  1. \({A}_i\Prime\vec{B}_i\Prime=A_i B_i\);
  2. If \(C_i\Prime=\epsilon_{ijk}A_j\Prime B_k\Prime\), then \(\vec{C}\Prime\) is given by an equation similar to \eqRef{EQ01}, where, of course, \(C_i=\epsilon_{ijk}A_j B_k\)
  3. The value of \( \vec{A}\Prime\cdot(\vec{B}\Prime\times\vec{C}\Prime)\) is independent of the matrix \(R\), if \(\det R=1\). What happens if \(\det R=-1\)?

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