https://file.aiquickdraw.com/mathbot/file/1725431718864ubgygw84.jpg

To solve the problem, we need to evaluate the expression \( 2 \cdot \vec{C} \cdot \left( \vec{A} \times \vec{B} \right) \), where the vectors are given as:

\[
\vec{A} = 3.00\hat{i} + 3.00\hat{j} - 4.00\hat{k}
\]
\[
\vec{B} = -3.00\hat{i} + 2.00\hat{j} + 2.00\hat{k}
\]
\[
\vec{C} = 7.00\hat{i} - 9.00\hat{j}
\]

### Step 1: Calculate the cross product \(\vec{A} \times \vec{B}\)

The cross product \(\vec{A} \times \vec{B}\) is computed using the determinant of a matrix composed of the unit vectors \(\hat{i}, \hat{j}, \hat{k}\) and the components of vectors \(\vec{A}\) and \(\vec{B}\):

\[
\vec{A} \times \vec{B} = \begin{vmatrix} 
\hat{i} & \hat{j} & \hat{k} \\ 
3.00 & 3.00 & -4.00 \\ 
-3.00 & 2.00 & 2.00 
\end{vmatrix}
\]

Expanding this determinant:

\[
\vec{A} \times \vec{B} = \hat{i}\left(3.00 \times 2.00 - (-4.00) \times 2.00\right) - \hat{j}\left(3.00 \times 2.00 - (-4.00) \times (-3.00)\right) + \hat{k}\left(3.00 \times 2.00 - 3.00 \times (-3.00)\right)
\]

Calculating each component:

\[
\vec{A} \times \vec{B} = \hat{i}(6.00 + 8.00) - \hat{j}(6.00 + 12.00) + \hat{k}(6.00 + 9.00)
\]

\[
\vec{A} \times \vec{B} = \hat{i}(14.00) - \hat{j}(18.00) + \hat{k}(15.00)
\]

\[
\vec{A} \times \vec{B} = 14.00\hat{i} - 18.00\hat{j} + 15.00\hat{k}
\]

### Step 2: Compute the dot product \(\vec{C} \cdot (\vec{A} \times \vec{B})\)

Now, the dot product of \(\vec{C}\) and \(\vec{A} \times \vec{B}\) is:

\[
\vec{C} \cdot (\vec{A} \times \vec{B}) = (7.00\hat{i} - 9.00\hat{j}) \cdot (14.00\hat{i} - 18.00\hat{j} + 15.00\hat{k})
\]

Only the \(\hat{i}\) and \(\hat{j}\) components contribute, since \(\vec{C}\) has no \(\hat{k}\) component:

\[
\vec{C} \cdot (\vec{A} \times \vec{B}) = 7.00 \times 14.00 + (-9.00) \times (-18.00)
\]

\[
\vec{C} \cdot (\vec{A} \times \vec{B}) = 98.00 + 162.00 = 260.00
\]

### Step 3: Calculate \( 2 \cdot \vec{C} \cdot (\vec{A} \times \vec{B}) \)

Finally, multiply the result by 2:

\[
2 \cdot 260.00 = 520.00
\]

### Final Answer:
\[
2 \cdot \vec{C} \cdot (\vec{A} \times \vec{B}) = 520.00
\]

Would you like more details or have any questions?

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Here are 5 related questions you might find useful:
1. How do you compute the magnitude of a cross product vector?
2. What is the geometric interpretation of the dot product between two vectors?
3. How does the cross product differ from the dot product in terms of the resulting vector or scalar?
4. Can the cross product of two parallel vectors be non-zero? Why or why not?
5. How do you determine the direction of the resultant vector from a cross product?

**Tip:** Remember that the cross product is anti-commutative, meaning \(\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})\).

For the following three vectors, what is 2 ⋅ C ⋅ (A × B)?

A = 3.00i + 3.00j - 4.00k
B = -3.00i + 2.00j + 2.00k
C = 7.00i - 9.00j

Solve the vector operation problem involving cross and dot products: For vectors A = 3.00i + 3.00j - 4.00k, B = -3.00i + 2.00j + 2.00k, and C = 7.00i - 9.00j, find 2 ⋅ C ⋅ (A × B) with a detailed step-by-step solution.

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