Question

If \( \mathbf{a} \) is a constant vector, then \( \text{div}[\mathbf{a} \times (\mathbf{r} \times \mathbf{a})] \) is

• A. \( 2a^2 \)
• B. \( 3a^2 \)
• C. \( 6a^2 \)
• D. \( 4a^2 \)

Hint:

In vector calculus, the divergence of a cross product can be simplified using the vector identity to find the result.

Answer 1

The Correct Option is A

We are asked to compute the divergence of the vector field \( \mathbf{a} \times (\mathbf{r} \times \mathbf{a}) \), where \( \mathbf{a} \) is a constant vector and \( \mathbf{r} \) is the position vector.

The vector triple product identity is given by:

\[ \mathbf{r} \times (\mathbf{a} \times \mathbf{b}) = (\mathbf{r} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{r} \cdot \mathbf{a}) \mathbf{b} \]

Substitute \( \mathbf{b} = \mathbf{a} \) into the identity to simplify the cross product \( \mathbf{r} \times (\mathbf{a} \times \mathbf{a}) \). Since the cross product of any vector with itself is zero, we get:

\[ \mathbf{r} \times (\mathbf{a} \times \mathbf{a}) = \mathbf{0} \]

Thus, we need to compute the divergence of the expression \( \mathbf{a} \times (\mathbf{r} \times \mathbf{a}) \). Using the vector identity for the divergence of a cross product:

\[ \nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) \]

For a constant vector \( \mathbf{a} \), the divergence simplifies to:

\[ \text{div}[\mathbf{a} \times (\mathbf{r} \times \mathbf{a})] = 2a^2 \]

The correct answer is option (A), \( 2a^2 \).