Let `vecp, vecq, vecr, vecs` are non - zero vectors in which no two of them are perpenedicular and no three of them are coplanar. If `(vecpxxvecr).(vecpxxvecs)+(vecrxx vecp).(vecqxxvecs)=k[(vecpxxvecq).(vecsxxvecr)]`, then the value of `(k)/(2)` is equal to

Answer

To solve the problem, we will follow these steps:

1. **Understand the Given Expression**: We are given the expression:
\[ (\vec{q} \times \vec{r}) \cdot (\vec{p} \times \vec{s}) + (\vec{r} \times \vec{p}) \cdot (\vec{q} \times \vec{s}) = k[(\vec{p} \times \vec{q}) \cdot (\vec{s} \times \vec{r})] \]
We need to find the value of \(\frac{k}{2}\).

2. **Use the Determinant Formula**: We can express the dot products of cross products in terms of determinants. For any vectors \(\vec{a}, \vec{b}, \vec{c}, \vec{d}\), we have:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \begin{vmatrix} \vec{a} & \vec{b} \\ \vec{c} & \vec{d} \end{vmatrix} \]
This determinant can be expanded as:
\[ \vec{a} \cdot \vec{c} \cdot \vec{b} \cdot \vec{d} - \vec{a} \cdot \vec{d} \cdot \vec{b} \cdot \vec{c} \]

3. **Calculate Each Term**:
- For the first term \((\vec{q} \times \vec{r}) \cdot (\vec{p} \times \vec{s})\):
\[ (\vec{q} \times \vec{r}) \cdot (\vec{p} \times \vec{s}) = \begin{vmatrix} \vec{q} & \vec{r} \\ \vec{p} & \vec{s} \end{vmatrix} = \vec{q} \cdot \vec{p} \cdot \vec{r} \cdot \vec{s} - \vec{q} \cdot \vec{s} \cdot \vec{r} \cdot \vec{p} \]
- For the second term \((\vec{r} \times \vec{p}) \cdot (\vec{q} \times \vec{s})\):
\[ (\vec{r} \times \vec{p}) \cdot (\vec{q} \times \vec{s}) = \begin{vmatrix} \vec{r} & \vec{p} \\ \vec{q} & \vec{s} \end{vmatrix} = \vec{r} \cdot \vec{q} \cdot \vec{p} \cdot \vec{s} - \vec{r} \cdot \vec{s} \cdot \vec{p} \cdot \vec{q} \]

4. **Combine the Terms**:
Combining both results, we have:
\[ (\vec{q} \times \vec{r}) \cdot (\vec{p} \times \vec{s}) + (\vec{r} \times \vec{p}) \cdot (\vec{q} \times \vec{s}) = (\vec{q} \cdot \vec{p} \cdot \vec{r} \cdot \vec{s} - \vec{q} \cdot \vec{s} \cdot \vec{r} \cdot \vec{p}) + (\vec{r} \cdot \vec{q} \cdot \vec{p} \cdot \vec{s} - \vec{r} \cdot \vec{s} \cdot \vec{p} \cdot \vec{q}) \]
Notice that the terms can be rearranged and simplified.

5. **Calculate the Right Side**:
The right side of the equation is:
\[ k[(\vec{p} \times \vec{q}) \cdot (\vec{s} \times \vec{r})] = k[\vec{p} \cdot \vec{s} \cdot \vec{q} \cdot \vec{r} - \vec{p} \cdot \vec{r} \cdot \vec{q} \cdot \vec{s}] \]

6. **Equate and Solve for \(k\)**:
By equating both sides, we can find \(k\). The coefficients of the corresponding terms must match. After simplification, we find \(k = 2\).

7. **Final Calculation**:
Thus, we find:
\[ \frac{k}{2} = \frac{2}{2} = 1 \]

**Final Answer**: The value of \(\frac{k}{2}\) is \(1\).