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We are tasked with solving the vector operation:

\[
2 \cdot \vec{C} \cdot (2\vec{A} \times \vec{B})
\]

### Given Vectors:

- \(\vec{A} = 3.00\hat{i} + 4.00\hat{j} - 3.00\hat{k}\)
- \(\vec{B} = -3.00\hat{i} + 3.00\hat{j} + 4.00\hat{k}\)
- \(\vec{C} = 7.00\hat{i} - 7.00\hat{j}\)

### Steps to Solve:

1. **Calculate \( 2\vec{A} \):**
   \[
   2\vec{A} = 2(3.00\hat{i} + 4.00\hat{j} - 3.00\hat{k}) = 6.00\hat{i} + 8.00\hat{j} - 6.00\hat{k}
   \]

2. **Compute the cross product \( 2\vec{A} \times \vec{B} \):**
   Use the determinant form of the cross product:

   \[
   2\vec{A} \times \vec{B} = 
   \begin{vmatrix}
   \hat{i} & \hat{j} & \hat{k} \\
   6.00 & 8.00 & -6.00 \\
   -3.00 & 3.00 & 4.00
   \end{vmatrix}
   \]

3. **Find the dot product \( \vec{C} \cdot (2\vec{A} \times \vec{B}) \):**
   After obtaining the result of the cross product, compute the dot product with \(\vec{C}\).

4. **Multiply by 2:**
   Finally, multiply the dot product by 2 to get the final result.

Let me compute this for you.The result of the vector operation \( 2 \cdot \vec{C} \cdot (2\vec{A} \times \vec{B}) \) is **784.0**.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How is the cross product between two vectors computed geometrically?
2. How does the dot product relate to the angle between two vectors?
3. What is the physical significance of the scalar triple product in physics?
4. Can the result of a vector cross product be a scalar?
5. How would the result change if \(\vec{C}\) had a non-zero \( \hat{k} \) component?

**Tip:** When computing cross products, the determinant of a matrix involving unit vectors can simplify the process.

For the following three vectors, what is 2 * C · (2A × B)?
A = 3.00i + 4.00j - 3.00k
B = -3.00i + 3.00j + 4.00k
C = 7.00i - 7.00j

This problem explores the calculation of 2C · (2A × B) using vector operations. Detailed steps include calculating the cross product of vectors A and B, and finding the dot product with vector C. Suitable for advanced high school or undergraduate students studying physics or vector algebra.

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