If `vecp=(vecbxxvecc)/([(veca,vecb,vecc)]),vecq=(veccxxveca)/([(veca,vecb,vecc)]),vecr=(vecaxxvecb)/([(veca,vecb,vecb)])` where `veca,vecb,vecc` are three non-coplanar vectors, then the value of the expression `(veca+vecb+vecc).(vecp+vecq+vecr)` is

Explanation

To solve the problem, we need to evaluate the expression \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{p} + \vec{q} + \vec{r})\) where:

\[ \vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a}, \vec{b}, \vec{c}]}, \quad \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a}, \vec{b}, \vec{c}]}, \quad \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a}, \vec{b}, \vec{c}]} \]

Here, \([\vec{a}, \vec{b}, \vec{c}]\) represents the scalar triple product of the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

### Step-by-Step Solution:

1. **Understanding Reciprocal Vectors**:
The vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) are known as reciprocal vectors of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). A property of reciprocal vectors states that:
\[ \vec{a} \cdot \vec{p} = 1, \quad \vec{b} \cdot \vec{q} = 1, \quad \vec{c} \cdot \vec{r} = 1 \]
and the dot products of any vector with the reciprocal of the others are zero:
\[ \vec{a} \cdot \vec{q} = 0, \quad \vec{a} \cdot \vec{r} = 0, \quad \vec{b} \cdot \vec{p} = 0, \quad \vec{b} \cdot \vec{r} = 0, \quad \vec{c} \cdot \vec{p} = 0, \quad \vec{c} \cdot \vec{q} = 0 \]

2. **Expanding the Expression**:
We need to evaluate:
\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{p} + \vec{q} + \vec{r}) \]
This can be expanded as:
\[ \vec{a} \cdot \vec{p} + \vec{a} \cdot \vec{q} + \vec{a} \cdot \vec{r} + \vec{b} \cdot \vec{p} + \vec{b} \cdot \vec{q} + \vec{b} \cdot \vec{r} + \vec{c} \cdot \vec{p} + \vec{c} \cdot \vec{q} + \vec{c} \cdot \vec{r} \]

3. **Applying the Properties**:
Using the properties of reciprocal vectors:
- \(\vec{a} \cdot \vec{p} = 1\)
- \(\vec{b} \cdot \vec{q} = 1\)
- \(\vec{c} \cdot \vec{r} = 1\)
- All other dot products are zero.

Thus, the expression simplifies to:
\[ 1 + 0 + 0 + 0 + 1 + 0 + 0 + 0 + 1 = 3 \]

4. **Final Result**:
Therefore, the value of the expression \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{p} + \vec{q} + \vec{r})\) is:
\[ \boxed{3} \]