If `vecl,vecm,vecn` are three non coplanar vectors prove that `[vec` vecm vecn](vecaxxvecb) =|(vec1.veca, vec1.vecb, vec1),(vecm.veca, vecm.vecb, vecm),(vecn.veca, vecn.vecb, vecn)|`

Explanation

To prove the given expression, we will start by analyzing both sides of the equation. The expression we need to prove is:

\[
[\vec{l}, \vec{m}, \vec{n}] (\vec{a} \times \vec{b}) = \left| \begin{array}{ccc}
\vec{l} \cdot \vec{a} & \vec{l} \cdot \vec{b} & \vec{l} \\
\vec{m} \cdot \vec{a} & \vec{m} \cdot \vec{b} & \vec{m} \\
\vec{n} \cdot \vec{a} & \vec{n} \cdot \vec{b} & \vec{n}
\end{array} \right|
...