If `veca, vecb, vecc` and `veca', vecb', vecc'` form a reciprocal system of vectors then
`veca.veca'+vecb.vecb'+vecc.vecc'=`

Options

A. `vec0`
B. `vecaxxb`
C. `vecbxxvecc`
D. `veccxxveca`

Explanation

To solve the problem, we need to find the value of the expression \( \vec{a} \cdot \vec{a}' + \vec{b} \cdot \vec{b}' + \vec{c} \cdot \vec{c}' \) given that \( \vec{a}, \vec{b}, \vec{c} \) and \( \vec{a}', \vec{b}', \vec{c}' \) form a reciprocal system of vectors.

### Step-by-Step Solution:

1. **Understanding Reciprocal Vectors**:
In a reciprocal system of vectors, each vector can be expressed in terms of the cross products of the other two vectors. Specifically:
\[ \vec{a}' = \frac{\vec{b} \times \vec{c}}{V}, \quad \vec{b}' = \frac{\vec{c} \times \vec{a}}{V}, \quad \vec{c}' = \frac{\vec{a} \times \vec{b}}{V} \]
where \( V = \vec{a} \cdot (\vec{b} \times \vec{c}) \) is the scalar triple product.

2. **Substituting the Expressions**:
Substitute the expressions for \( \vec{a}', \vec{b}', \vec{c}' \) into the given equation:
\[ \vec{a} \cdot \vec{a}' + \vec{b} \cdot \vec{b}' + \vec{c} \cdot \vec{c}' = \vec{a} \cdot \left(\frac{\vec{b} \times \vec{c}}{V}\right) + \vec{b} \cdot \left(\frac{\vec{c} \times \vec{a}}{V}\right) + \vec{c} \cdot \left(\frac{\vec{a} \times \vec{b}}{V}\right) \]

3. **Using the Scalar Triple Product**:
The dot product of a vector with a cross product can be expressed as a scalar triple product:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = V, \quad \vec{b} \cdot (\vec{c} \times \vec{a}) = V, \quad \vec{c} \cdot (\vec{a} \times \vec{b}) = V \]

4. **Calculating Each Term**:
Therefore, we can rewrite the expression:
\[ \vec{a} \cdot \vec{a}' = \frac{V}{V} = 1, \quad \vec{b} \cdot \vec{b}' = \frac{V}{V} = 1, \quad \vec{c} \cdot \vec{c}' = \frac{V}{V} = 1 \]

5. **Summing the Results**:
Now, we can sum these results:
\[ \vec{a} \cdot \vec{a}' + \vec{b} \cdot \vec{b}' + \vec{c} \cdot \vec{c}' = 1 + 1 + 1 = 3 \]

### Final Answer:
Thus, the value of \( \vec{a} \cdot \vec{a}' + \vec{b} \cdot \vec{b}' + \vec{c} \cdot \vec{c}' \) is \( \boxed{3} \).