If `vec(b)` and `vec(c)` are two non-collinear vectors such that `vec(a) || (vec(b) xx vec(c))` , then `(vec(a) xx vec(b)).(vec(a) xx vec(c))` is equal to

Options

A. `vec(a)^(2) (vec(b).vec(c))`
B. `vec(b)^(2)(vec(a).vec(c))`
C. `vec(c)^(2) (vec(a).vec(b))`
D. none of these

Answer

To solve the problem, we need to find the value of \((\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c})\) given that \(\vec{a} || (\vec{b} \times \vec{c})\).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
Since \(\vec{a} || (\vec{b} \times \vec{c})\), it implies that \(\vec{a}\) is perpendicular to both \(\vec{b}\) and \(\vec{c}\). This is because the cross product \(\vec{b} \times \vec{c}\) is perpendicular to the plane formed by \(\vec{b}\) and \(\vec{c}\).

2. **Using the Dot Product**:
From the property of the dot product, if two vectors are perpendicular, their dot product is zero. Therefore:
\[ \vec{a} \cdot \vec{b} = 0 \quad \text{and} \quad \vec{a} \cdot \vec{c} = 0 \]

3. **Applying the Scalar Triple Product**:
The expression \((\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c})\) can be rewritten using the scalar triple product:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c}) = \vec{b} \cdot (\vec{a} \times \vec{c}) \times \vec{a} \]
This is equivalent to the scalar triple product \([\vec{a}, \vec{b}, \vec{c}]\).

4. **Using the Vector Triple Product Identity**:
We can use the vector triple product identity:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
Applying this to our case:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

5. **Substituting the Dot Products**:
Since we established that \(\vec{a} \cdot \vec{b} = 0\) and \(\vec{a} \cdot \vec{c} = 0\), we substitute these into the equation:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = 0 \cdot \vec{b} - 0 \cdot \vec{c} = \vec{0} \]

6. **Final Result**:
Since \((\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c})\) simplifies to zero, we conclude:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c}) = 0 \]

### Conclusion:
Thus, the value of \((\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c})\) is **0**.