<s>[INST] Show that $(\mathbf{a}·\mathbf{a})(\mathbf{b}·\mathbf{b})-(\mathbf{a}·\mathbf{b})^2=(\mathbf{a}\times\mathbf{b})·(\mathbf{a}\times\mathbf{b})$? I need to prove that, for two linearly independent vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^3$,
$$(\mathbf{a} · \mathbf{a}) (\mathbf{b} · \mathbf{b}) - (\mathbf{a} · \mathbf{b})(\mathbf{a} · \mathbf{b}) = (\mathbf{a} \times \mathbf{b})·(\mathbf{a} \times \mathbf{b})$$
Could someone give me a demonstration of this identity? Or a hint to prove it?
 [/INST] The left hand side is the determinant of the Gramian matrix of $\mathbf a$ and $\mathbf b$:
$$
\det\begin{pmatrix}\mathbf a\cdot\mathbf a & \mathbf a\cdot\mathbf b \\
\mathbf b\cdot\mathbf a & \mathbf b\cdot\mathbf b\end{pmatrix}.
$$
In general, the determinant of a Gramian matrix is the square of the $n$-dimensional volume of a parallelotope, in this case the square of the area of the parallelogram spanned by $\mathbf a$ and $\mathbf b$.
The right hand side is $\|\mathbf a\times\mathbf b\|^2$, where $\|\mathbf a\times\mathbf b\|$ also is the area of the parallelogram spanned by $\mathbf a$ and $\mathbf b$, hence both sides are equal.
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