Let `veca,vecb and vecc` be a set of non- coplanar vectors and `veca'vecb' and vecc'` be its reciprocal set.
prove that `veca=(vecb'xxvecc')/([veca'vecb'vecc']),vecb=(vecc'xxveca')/([veca'vecb'vecc'])andvecc=(veca'xxvecb')/([veca'vecb'vecc'])`

Explanation

To prove the relationships between the vectors \( \vec{a}, \vec{b}, \vec{c} \) and their reciprocal set \( \vec{a}', \vec{b}', \vec{c}' \), we will follow a structured approach based on the properties of vector products and scalar triple products.

### Step-by-Step Solution:

1. **Understanding the Reciprocal Set**:
The reciprocal set of vectors \( \vec{a}', \vec{b}', \vec{c}' \) is defined as follows:
\[
\vec{a}' = \frac{\vec{b} \times \vec{c}}{[\vec{a}, \vec{b}, \vec{c}]}, \quad \vec{b}' = \frac{\vec{c} \times \vec{a}}{[\vec{a}, \vec{b}, \vec{c}]}, \quad \vec{c}' = \frac{\vec{a} \times \vec{b}}{[\vec{a}, \vec{b}, \vec{c}]}
...