If `veca,vecb,vecc` be three vectors such that `[veca vecb vec c]=4` then `[vecaxxvecb vecbxxvecc veccxxveca]` is equal to

Options

A. 8
B. 64
C. 4
D. 16

Answer

To solve the problem, we need to find the value of the scalar triple product \([ \vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a} ]\) given that \([ \vec{a}, \vec{b}, \vec{c} ] = 4\).

### Step-by-step Solution:

1. **Understanding the Given Information**:
We know that the scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\) is defined as the volume of the parallelepiped formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Given that this value is \(4\), we have:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = 4 \]

2. **Setting Up the Expression**:
We need to evaluate the scalar triple product:
\[ [ \vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a} ] \]

3. **Using the Vector Triple Product Identity**:
We can use the vector triple product identity which states:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
Applying this identity, we can express:
\[ [ \vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a} ] = \vec{a} \times \vec{b} \cdot (\vec{b} \times \vec{c} \times \vec{c} \times \vec{a}) \]

4. **Evaluating the Inner Product**:
We can rewrite the expression:
\[ \vec{b} \times \vec{c} \times \vec{c} \times \vec{a} = \vec{b} \times \vec{c} \cdot \vec{a} = [\vec{b}, \vec{c}, \vec{a}] \]
Since \(\vec{b} \times \vec{c}\) is perpendicular to \(\vec{c}\), the dot product \(\vec{b} \times \vec{c} \cdot \vec{c}\) is zero.

5. **Final Calculation**:
Thus, we have:
\[ [ \vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a} ] = [\vec{a}, \vec{b}, \vec{c}]^2 \]
Substituting the known value:
\[ [ \vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a} ] = 4^2 = 16 \]

### Conclusion:
The value of \([ \vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a} ]\) is \(16\).