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- #1

With respect to the triad of base vectors $\mathbf{u}_1$, $\mathbf{u}_2$, and $\mathbf{u}_3$ (not necessary unit vectors), the triad $\mathbf{u}^1$, $\mathbf{u}^2$, and $\mathbf{u}^3$ is said to be the reciprocal basis if $\mathbf{u}_i\cdot\mathbf{u}^j = \delta_{ij}$.

Show that to satisfy these conditions

$$

\mathbf{u}^1 = \frac{\mathbf{u}_3\times\mathbf{u}_1}{[\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3]};\quad

\mathbf{u}^2 = \frac{\mathbf{u}_2\times\mathbf{u}_3}{[\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3]};\quad

\mathbf{u}^3 = \frac{\mathbf{u}_1\times\mathbf{u}_2}{[\mathbf{u}_1,\mathbf{u}_2,\mathbf{u}_3]}

$$

and determine the reciprocal basis for the specific base vectors

\begin{alignat*}{3}

\mathbf{u}_1 & = & 2\hat{\mathbf{e}}_1 + \hat{\mathbf{e}}_2,\\

\mathbf{u}_2 & = & 2\hat{\mathbf{e}}_2 - \hat{\mathbf{e}}_3,\\

\mathbf{u}_3 & = & \hat{\mathbf{e}}_1 + \hat{\mathbf{e}}_2 + \hat{\mathbf{e}}_3

\end{alignat*}