Question

If $\overline{a}$ is constant vector then $\nabla \times (\overline{a} \times \overline{r}) = _?$

• A. $\overline{a}$
• B. $2\overline{a}$
• C. $3\overline{a}$
• D. $\overline{0}$

Hint:

For constant $\overline{a}$, the curl of $(\overline{a} \times \overline{r})$ is always $2\overline{a}$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the vector identity for the curl of a cross product: $\nabla \times (\overline{A} \times \overline{B}) = (\overline{B} \cdot \nabla)\overline{A} - (\overline{A} \cdot \nabla)\overline{B} + \overline{A}(\nabla \cdot \overline{B}) - \overline{B}(\nabla \cdot \overline{A})$.

Step 2: Meaning
Let $\overline{A} = \overline{a}$ (constant) and $\overline{B} = \overline{r} = xi + yj + zk$. Since $\overline{a}$ is constant, its derivatives $(\overline{r} \cdot \nabla)\overline{a}$ and $\nabla \cdot \overline{a}$ are zero.

Step 3: Analysis
The expression reduces to: $\overline{a}(\nabla \cdot \overline{r}) - (\overline{a} \cdot \nabla)\overline{r}$. We know $\nabla \cdot \overline{r} = 3$ and $(\overline{a} \cdot \nabla)\overline{r} = \overline{a}$.

Step 4: Conclusion
Calculation: $3\overline{a} - \overline{a} = 2\overline{a}$. Final Answer: (B)