Question

If \[ \overline{a}\ \text{and}\ \overline{b} \] are constant vectors and \[ \overline{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}, \] then \[ \nabla\!\left(\overline{r}\cdot(\overline{a}\times\overline{b})\right)=\_ \]

• A. $\overline{a} \cdot \overline{b}$
• B. $\overline{a} \times \overline{b}$
• C. $\overline{b} \times \overline{a}$
• D. 0

Hint:

The gradient of the dot product of the position vector $\overline{r}$ with any constant vector is just that constant vector.

Answer 1

The Correct Option is B

Step 1: Concept
Use the property $\nabla(\overline{r} \cdot \overline{c}) = \overline{c}$ where $\overline{c}$ is a constant vector.

Step 2: Meaning
In the expression $\nabla(\overline{r} \cdot (\overline{a} \times \overline{b}))$, let $\overline{c} = \overline{a} \times \overline{b}$. Since $\overline{a}$ and $\overline{b}$ are constant, their cross product $\overline{c}$ is also a constant vector.

Step 3: Analysis
Applying the rule: $\nabla(\overline{r} \cdot \overline{c}) = \overline{c}$.

Step 4: Conclusion
Substituting back: $\nabla(\overline{r} \cdot (\overline{a} \times \overline{b})) = \overline{a} \times \overline{b}$. Final Answer: (B)