If `veca, vecb, vecc` are non coplanar vectors and `vecp, vecq, vecr` are reciprocal vectors, then
`(lveca+mvecb+nvecc).(lvecp+mvecq+nvecr)` is equal to

Options

A. `l^(2)+m^(2)+n^(2)`
B. lm+mn+nl
C. 0
D. None of these

Explanation

To solve the problem, we need to evaluate the expression \((l\vec{a} + m\vec{b} + n\vec{c}) \cdot (l\vec{p} + m\vec{q} + n\vec{r})\), where \(\vec{a}, \vec{b}, \vec{c}\) are non-coplanar vectors and \(\vec{p}, \vec{q}, \vec{r}\) are reciprocal vectors.

### Step-by-step Solution:

1. **Understanding Reciprocal Vectors**:
- The reciprocal vectors can be expressed as:
\[ \vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a}, \vec{b}, \vec{c}]}, \quad \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a}, \vec{b}, \vec{c}]}, \quad \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a}, \vec{b}, \vec{c}]} \]
- Here, \([\vec{a}, \vec{b}, \vec{c}]\) denotes the scalar triple product, which is non-zero since the vectors are non-coplanar.

2. **Expanding the Dot Product**:
- We expand the expression:
\[ (l\vec{a} + m\vec{b} + n\vec{c}) \cdot (l\vec{p} + m\vec{q} + n\vec{r}) \]
- This can be expanded as:
\[ l^2 (\vec{a} \cdot \vec{p}) + lm (\vec{a} \cdot \vec{q}) + ln (\vec{a} \cdot \vec{r}) + ml (\vec{b} \cdot \vec{p}) + m^2 (\vec{b} \cdot \vec{q}) + mn (\vec{b} \cdot \vec{r}) + nl (\vec{c} \cdot \vec{p}) + nm (\vec{c} \cdot \vec{q}) + n^2 (\vec{c} \cdot \vec{r}) \]

3. **Evaluating Each Dot Product**:
- Using the properties of reciprocal vectors:
- \(\vec{a} \cdot \vec{p} = 1\)
- \(\vec{a} \cdot \vec{q} = 0\)
- \(\vec{a} \cdot \vec{r} = 0\)
- \(\vec{b} \cdot \vec{p} = 0\)
- \(\vec{b} \cdot \vec{q} = 1\)
- \(\vec{b} \cdot \vec{r} = 0\)
- \(\vec{c} \cdot \vec{p} = 0\)
- \(\vec{c} \cdot \vec{q} = 0\)
- \(\vec{c} \cdot \vec{r} = 1\)

4. **Substituting the Values**:
- Substitute the values into the expanded expression:
\[ l^2(1) + lm(0) + ln(0) + ml(0) + m^2(1) + mn(0) + nl(0) + nm(0) + n^2(1) \]
- This simplifies to:
\[ l^2 + m^2 + n^2 \]

5. **Final Result**:
- Therefore, the value of \((l\vec{a} + m\vec{b} + n\vec{c}) \cdot (l\vec{p} + m\vec{q} + n\vec{r})\) is:
\[ l^2 + m^2 + n^2 \]

### Final Answer:
\[ \boxed{l^2 + m^2 + n^2} \]