Exercise:
In this problem we consider some useful identities involving the cross-product of two thredimensional vectors pmatrix a b c pmatrix times pmatrix x y z pmatrix : pmatrix bz-cy cx-az ay-bx pmatrix and the Levi-Civita symbol epsilon^ijk with ijkin . The Levi-Civita symbol is + for even permutations of - for odd permutations of and otherwise. abcliste abc Show that the cross product can be written in terms of epsilon^ijk as vecatimes vecb^k _ij^ epsilon^ijka^ib^j. abc Show the following identities itemize item veca times vecb vecc vecbtimes vecc veca vecctimes veca vecb item vecatimes vecbtimes vecc veca veccvecb-vecb veccveca item vecatimes vecb vecctimes vecd vecd vecatimes vecbtimes vecc veca veccvecb vecd-vecb veccveca vecd itemize abc Show that for a times matrix M _lmn^ epsilon^lmnM^liM^mjM^nkepsilon^ijktextdetM. abc Show that under rotations Rin textSO Rvecatimes Rvecb Rvecatimes vecb. How can this identity be generalised for Rin textO? abc Find a times matrix Omega such that vecomegatimes vecaOmega veca for all vectors veca when vecomega is a constant vector. What property does the matrix Omega have? abcliste

Solution:
abcliste abc The first component can be written as _ijepsilon^ija^ib^j epsilon^a^b^+epsilon^a^b^ a^b^-a^b^ vecatimes vecb^ The other components are obtained similarly. abc itemize item vecatimes vecb vecc _ijkepsilon^ijka^ib^jc^k _ijkepsilon^kijc^ka^ib^j vecctimes veca vecb _ijkepsilon^jkib^jc^ka^i vecbtimes vecc veca. item vecatimes vecbtimes vecc^k _ijepsilon^ijkvecatimes vecb^ic^j c^ka^ib^j _ijkmepsilon^ijkepsilon^imna^mb^nc^j. Notice that epsilon^ijkepsilon^imn cases quad textif ineq jneq k textand jm kn -quad textif ineq jneq k textand jn km quad textotherwisecases delta^jmdelta^kn-delta^jndelta^km cases quad textif jm kn jneq k -quad textif jn km jneq k quad textotherwisecases Therefore _i epsilon^ijk epsilon^imn delta^jmdelta^kn-delta^jndelta^km. Plugging this o we obtain vecatimes vecbtimes vecc^k _jmndelta^jmdelta^kn-delta^jndelta^kma^mb^nc^j _j a^jc^jb^k-b^jc^ja^k veca veccb^k-vecb vecca^k item Using the previous results: vecatimes vecb vecctimes vecd vecd vecatimes vecbtimes vecc veca veccvecb vecd-vecb veccveca vecd. itemize abc Let vecm_i be the i-th column of M. Then textdetM vecm_ vecm_times vecm_. From the definition of the cross product we can easily see vecatimes vecb -vecbtimes veca and vecatimes veca. Applying the result brom before we can write epsilon^ijktextdetM vecm_i vecm_jtimes vecm_k _lmn epsilon^lmnM^liM^mjM^nk. abc Let R be an invertible matrix. R^TRvecatimes Rvecb^i _j R^jiRvecatimes Rvecb^j _jklepsilon^jklR^ijRveca^kRvecb^l _jklmnepsilon^jklR^jiR^kmR^lna^mb^n _mnepsilon^imntextdetRa^mb^n textdetRvecatimes vecb^i. Therefore Rvecatimes RvecbtextdetRR^T^-vecatimes vecb for an invertible R. when Rin textO R^T^-R and Rvecatimes RvecbtextdetRRvecatimes vecb. When Rin textSO textdetR and this simplifies to the in the ning. abc For Omega pmatrix & -omega^ & omega^ omega^ & & -omega^ -omega^ & omega^ & pmatrix we get Omega veca pmatrix omega^a^-omega^a^ omega^a^-omega^a^ omega^a^-omega^a^ pmatrix omega times veca. abcliste