Prove, using the standard formula connecting $\delta_{i j}$ and $\epsilon_{i j k}$, that

$$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$$

Define, in terms of the dot and cross product, the triple scalar product [a, b, c $]$ of three vectors $\mathbf{a}, \mathbf{b}, \mathbf{c}$ in $\mathbb{R}^{3}$ and show that it is invariant under cyclic permutation of the vectors.

Let $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ be a not necessarily orthonormal basis for $\mathbb{R}^{3}$, and define

$$\hat{\mathbf{e}}_{1}=\frac{\mathbf{e}_{2} \times \mathbf{e}_{3}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}, \quad \hat{\mathbf{e}}_{2}=\frac{\mathbf{e}_{3} \times \mathbf{e}_{1}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]}, \quad \hat{\mathbf{e}}_{3}=\frac{\mathbf{e}_{1} \times \mathbf{e}_{2}}{\left[\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}\right]} .$$

By calculating $\left[\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}\right]$, show that $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ is also a basis for $\mathbb{R}^{3}$.

The vectors $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ are constructed from $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ in the same way that $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ are constructed from $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. Show that

$$\hat{\mathbf{e}}_{1}=\mathbf{e}_{1}, \hat{\hat{\mathbf{e}}}_{2}=\mathbf{e}_{2}, \hat{\mathbf{e}}_{3}=\mathbf{e}_{3},$$

Show that a vector $\mathbf{V}$ has components $\mathbf{V} \cdot \hat{\mathbf{e}}_{1}, \mathbf{V} \cdot \hat{\mathbf{e}}_{2}, \mathbf{V} \cdot \hat{\mathbf{e}}_{3}$ with respect to the basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. What are the components of the vector $\mathbf{V}$ with respect to the basis $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ ?