If `vecp = (vecbxxvecc)/([vecavecbvecc]), vecq=(veccxxveca)/([veca vecb vecc])and vecr = (vecaxxvecb)/([veca vecbvecc]), " where " veca,vecb and vecc` are three non- coplanar vectors then the value of the expression `(veca + vecb + vecc ). (vecp+ vecq+vecr)` is `(a)3` `(b)2` `(c)1` `(d)0`

Options

A. 3
B. 2
C. 1
D. 0

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 1: Understand the Properties of Reciprocal Vectors
The vectors \(\vec{p}\), \(\vec{q}\), and \(\vec{r}\) are known as the reciprocal vectors of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). A property of these reciprocal vectors is 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 the reciprocal vectors with the original vectors yield zero:

\[ \vec{a} \cdot \vec{q} = 0, \quad \vec{a} \cdot \vec{r} = 0, \quad \vec{b} \cdot \vec{p} = 0, \quad \vec{b} \cdot \vec{r} = 0, \quad \vec{c} \cdot \vec{p} = 0, \quad \vec{c} \cdot \vec{q} = 0 \]

### Step 2: Evaluate the Expression
Now, we can evaluate the expression:

\[ (\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{p} + \vec{q} + \vec{r}) \]

Expanding this using the distributive property of the dot product, we have:

\[ = \vec{a} \cdot \vec{p} + \vec{a} \cdot \vec{q} + \vec{a} \cdot \vec{r} + \vec{b} \cdot \vec{p} + \vec{b} \cdot \vec{q} + \vec{b} \cdot \vec{r} + \vec{c} \cdot \vec{p} + \vec{c} \cdot \vec{q} + \vec{c} \cdot \vec{r} \]

Using the properties of the reciprocal vectors, we substitute:

\[ = 1 + 0 + 0 + 0 + 1 + 0 + 0 + 0 + 1 \]

### Step 3: Simplify the Expression
Now, simplifying the above expression gives:

\[ = 1 + 1 + 1 = 3 \]

### Conclusion
Thus, the value of the expression \((\vec{a} + \vec{b} + \vec{c}) \cdot (\vec{p} + \vec{q} + \vec{r})\) is:

\[ \boxed{3} \]