[QUE/ME-02007] ME-PROBLEM

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 \(\vec{A}{'}=(A_1{'},A_2{'},A_3{'}), \vec{B}{'}=(B_1{'},B_2{'},B_3{'})\) etc. by means of equation \begin{equation}\label{EQ01} \begin{pmatrix}A_1{'}\\A_2{'}\\A_3{'}\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

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