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Using indicial notation, I am trying to show that $\mathbf{v}\cdot\mathbf{v} = a^2b^2\sin^2\theta$ where $ \mathbf{v} = \mathbf{a}\times\mathbf{b}$ and $\mathbf{v}_i\hat{\mathbf{e}}_i = a_j \hat{\mathbf{e}}_j\times b_j\hat{\mathbf{e}}_k = \varepsilon_{ijk}a_jb_k\hat{\mathbf{e}}_i$.

So

\begin{alignat}{3}

\mathbf{v}\cdot\mathbf{v} & = & \varepsilon_{ijk} a_jb_k\hat{\mathbf{e}}_i\cdot\varepsilon_{ijk}a_jb_k\hat{\mathbf{e}}_i\\

& = &

\end{alignat}

We have to have a kronecker delta since the only surviving terms are when the unit vectors that are dotted with themselves but that is all I have.

So

\begin{alignat}{3}

\mathbf{v}\cdot\mathbf{v} & = & \varepsilon_{ijk} a_jb_k\hat{\mathbf{e}}_i\cdot\varepsilon_{ijk}a_jb_k\hat{\mathbf{e}}_i\\

& = &

\end{alignat}

We have to have a kronecker delta since the only surviving terms are when the unit vectors that are dotted with themselves but that is all I have.

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