(a) Let (ê, ê, ê3) be unit vectors of a right-handed, orthogonal coordinate system. Show that theLevi-Cività symbol satisfies \epsilon_{i j k}=\hat{\mathbf{e}}_{i} \cdot\left(\hat{\mathbf{e}}_{j} \times \hat{e}_{k}\right) (b) Prove that \epsilon_{i j l} \epsilon_{i s t}=\delta_{j s} \delta_{l t}-\delta_{j t} \delta_{l a} given that \epsilon_{j j k} \epsilon_{r a t}=\left|\begin{array}{lll}

\delta_{i r} & \delta_{i s} & \delta_{j t} \\

\delta_{j r} & \delta_{j s} & \delta_{j t} \\

\delta_{k r} & \delta_{k s} & \delta_{l t}

\end{array}\right| (c) In quantum mechanics the cartesian components of the angular momentum operator L obey commutation relation [ο‚Îj] = iħ€¿jkÎk. Let a and b be constant vectors and prove the com-mutator identity \left[\hat{\mathbf{L}} \cdot \mathbf{a}_{,} \hat{\mathbf{L}} \cdot \mathbf{b}\right]=\mathbf{i} \hat{\mathbf{L}} \cdot(\mathbf{a} \times \mathbf{b}) (d) Prove that a \times b=\epsilon_{j i b} \hat{\hat{\epsilon}}_{j} b_{j} b_{k}