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This question involves vector calculus and asks us to evaluate \(\nabla \times (\vec{r} \times \vec{F})\), where:

- \(\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}\), the position vector.
- \(\vec{F}\) is a general three-dimensional vector field.

### Formula Used:
We use the **vector triple product expansion** for \(\nabla \times (\vec{r} \times \vec{F})\). The formula is:

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

Here:
- \(\vec{A} = \vec{r}\),
- \(\vec{B} = \vec{F}\).

### Applying the Formula:
Breaking it into terms:

1. **First term**: \((\vec{F} \cdot \nabla) \vec{r}\):
   - Since \(\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}\),
   - \((\vec{F} \cdot \nabla) \vec{r} = \vec{F}\) (as differentiation gives back components of \(\vec{F}\)).

2. **Second term**: \((\vec{r} \cdot \nabla) \vec{F}\):
   - This term represents the directional derivative of \(\vec{F}\) along \(\vec{r}\).

3. **Third term**: \(\vec{r} (\nabla \cdot \vec{F})\):
   - \(\nabla \cdot \vec{F}\) is the divergence of \(\vec{F}\), so \(\vec{r} (\nabla \cdot \vec{F})\) scales \(\vec{r}\) by that value.

4. **Fourth term**: \(- \vec{F} (\nabla \cdot \vec{r})\):
   - \(\nabla \cdot \vec{r} = 3\) (since \(\partial x/\partial x + \partial y/\partial y + \partial z/\partial z = 1 + 1 + 1 = 3\)).
   - So, this term becomes \(-3 \vec{F}\).

### Combining the Results:
\[
\nabla \times (\vec{r} \times \vec{F}) = (\nabla \cdot \vec{F}) \vec{r} - (\vec{r} \cdot \nabla) \vec{F} - 3 \vec{F}
\]

This matches **option (d)**:
\[
(\nabla \cdot \vec{F}) \vec{r} - (\vec{r} \cdot \nabla) \vec{F} - 3 \vec{F}.
\]

### Final Answer:
**(d)**

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Let me know if you'd like a more detailed breakdown or examples! Here are five related questions to expand your understanding:

1. What is the physical significance of the divergence of a vector field?
2. How does the vector triple product formula generalize for various vector fields?
3. Can you derive \(\nabla \times (\nabla \times \vec{F})\) using similar principles?
4. What is the importance of the position vector \(\vec{r}\) in vector calculus?
5. How can the curl of a vector field be interpreted geometrically?

**Tip**: Always remember the identity \(\nabla \cdot \vec{r} = 3\) in three dimensions—it’s a useful shortcut!

If F is a three-dimensional vector field and r = xi + yj + zk, then ∇ × (r × F) will be?

This vector calculus problem evaluates ∇ × (r × F), where r = xi + yj + zk and F is a three-dimensional vector field. Using the vector triple product expansion, the solution involves curl, divergence, and directional derivatives. Suitable for undergraduate students.

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