https://file.aiquickdraw.com/mathbot/file/1728998988060a6gtsp24.jpg

The problem is asking to calculate the value of \( 3\vec{C} \cdot (\vec{A} \times \vec{B}) \), where the vectors are:

\[
\vec{A} = 2\hat{i} + 3\hat{j} - 4\hat{k}, \quad \vec{B} = -3\hat{i} + 4\hat{j} - 2\hat{k}, \quad \vec{C} = 7\hat{i} - 8\hat{j}.
\]

### Step 1: Find \( \vec{A} \times \vec{B} \) (the cross product)
The cross product of two vectors \( \vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \) is given by the determinant:

\[
\vec{A} \times \vec{B} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3
\end{vmatrix}
= \hat{i} \left( a_2b_3 - a_3b_2 \right) - \hat{j} \left( a_1b_3 - a_3b_1 \right) + \hat{k} \left( a_1b_2 - a_2b_1 \right)
\]

Substituting the components of \( \vec{A} \) and \( \vec{B} \):

\[
\vec{A} = \begin{pmatrix} 2 \\ 3 \\ -4 \end{pmatrix}, \quad \vec{B} = \begin{pmatrix} -3 \\ 4 \\ -2 \end{pmatrix}
\]

So,

\[
\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -4 \\ -3 & 4 & -2 \end{vmatrix}
\]

Expanding the determinant:

\[
\vec{A} \times \vec{B} = \hat{i} \left( 3 \times (-2) - (-4) \times 4 \right) - \hat{j} \left( 2 \times (-2) - (-4) \times (-3) \right) + \hat{k} \left( 2 \times 4 - 3 \times (-3) \right)
\]

Calculating the components:

- \( \hat{i} \) component: \( 3 \times (-2) - (-4) \times 4 = -6 + 16 = 10 \)
- \( \hat{j} \) component: \( 2 \times (-2) - (-4) \times (-3) = -4 - 12 = -16 \)
- \( \hat{k} \) component: \( 2 \times 4 - 3 \times (-3) = 8 + 9 = 17 \)

Thus, the cross product is:

\[
\vec{A} \times \vec{B} = 10\hat{i} - 16\hat{j} + 17\hat{k}
\]

### Step 2: Find \( 3\vec{C} \)
Next, we scale vector \( \vec{C} \) by 3:

\[
3\vec{C} = 3(7\hat{i} - 8\hat{j}) = 21\hat{i} - 24\hat{j}
\]

### Step 3: Find the dot product \( 3\vec{C} \cdot (\vec{A} \times \vec{B}) \)
Now, calculate the dot product between \( 3\vec{C} \) and \( \vec{A} \times \vec{B} \):

\[
3\vec{C} \cdot (\vec{A} \times \vec{B}) = (21\hat{i} - 24\hat{j}) \cdot (10\hat{i} - 16\hat{j} + 17\hat{k})
\]

This dot product is:

\[
= 21 \times 10 + (-24) \times (-16) + (0 \times 17)
\]
\[
= 210 + 384 + 0 = 594
\]

### Final Answer:
The value of \( 3\vec{C} \cdot (\vec{A} \times \vec{B}) \) is **594**.

Would you like further details or clarifications?

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Here are some related questions:
1. How is the cross product of two vectors different from the dot product?
2. What are the geometric interpretations of the dot and cross products?
3. How would the calculation change if one of the vectors was in 2D instead of 3D?
4. Can the result of a cross product ever be zero? If so, under what conditions?
5. How is the magnitude of the cross product related to the area of a parallelogram?

**Tip**: Always keep track of signs when dealing with cross products, as they directly affect the direction of the resulting vector.

Untuk tiga vektor berikut, berapakah nilai dari 3\vec{C} \cdot (2\vec{A} \times \vec{B})?

This solution demonstrates how to compute 3\vec{C} \cdot (2\vec{A} \times \vec{B}) using cross and dot products. The vectors provided are A = 2\hat{i} + 3\hat{j} - 4\hat{k}, B = -3\hat{i} + 4\hat{j} - 2\hat{k}, and C = 7\hat{i} - 8\hat{j}. Follow the step-by-step process to solve this problem and learn about vector operations in detail.

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