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The question asks to find the value of \( 3 \vec{C} \cdot (2 \vec{A} \times \vec{B}) \) given the following vectors:

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

To solve this, we will follow these steps:

### Step 1: Compute the cross product \( \vec{A} \times \vec{B} \)
The cross product of two vectors \( \vec{A} \times \vec{B} \) is given by the determinant of the following matrix:

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

This can be expanded as:

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

Now, compute each 2x2 determinant:

\[
\hat{i}: \begin{vmatrix} 3 & -4 \\ 4 & -2 \end{vmatrix} = (3)(-2) - (4)(-4) = -6 + 16 = 10
\]
\[
\hat{j}: \begin{vmatrix} 2 & -4 \\ -3 & -2 \end{vmatrix} = (2)(-2) - (-4)(-3) = -4 - 12 = -16
\]
\[
\hat{k}: \begin{vmatrix} 2 & 3 \\ -3 & 4 \end{vmatrix} = (2)(4) - (3)(-3) = 8 + 9 = 17
\]

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

### Step 2: Multiply by 2 to get \( 2 \vec{A} \times \vec{B} \)

\[
2 \vec{A} \times \vec{B} = 2(10 \hat{i} + 16 \hat{j} + 17 \hat{k}) = 20 \hat{i} + 32 \hat{j} + 34 \hat{k}
\]

### Step 3: Compute the dot product \( 3 \vec{C} \cdot (2 \vec{A} \times \vec{B}) \)

First, scale \( \vec{C} \) by 3:

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

Now, compute the dot product between \( 3 \vec{C} \) and \( 2 \vec{A} \times \vec{B} \):

\[
3 \vec{C} \cdot (2 \vec{A} \times \vec{B}) = (21 \hat{i} - 24 \hat{j}) \cdot (20 \hat{i} + 32 \hat{j} + 34 \hat{k})
\]

This simplifies to:

\[
= 21(20) + (-24)(32) + 0(34) = 420 - 768 = -348
\]

Thus, the value of \( 3 \vec{C} \cdot (2 \vec{A} \times \vec{B}) \) is **-348**.

Would you like further clarification on any step?

---

Here are 5 related questions you might find helpful:
1. How do you find the magnitude of a cross product between two vectors?
2. What is the geometric meaning of the dot product and the cross product?
3. Can a cross product between two vectors result in a scalar?
4. How can we interpret a negative dot product in terms of vector angles?
5. What is the difference between vector and scalar multiplication in vector algebra?

**Tip**: Always check if a vector operation requires the cross product or dot product, as their meanings and results are distinct in vector algebra.

Untuk tiga vektor berikut, berapakah nilai dari 3\vec{C} \cdot (2 \vec{A} \times \vec{B})? Diberikan vektor \vec{A} = 2\hat{i} + 3\hat{j} - 4\hat{k}, \vec{B} = -3\hat{i} + 4\hat{j} - 2\hat{k}, \vec{C} = 7\hat{i} - 8\hat{j}.

This problem involves calculating the value of 3\vec{C} \cdot (2 \vec{A} \times \vec{B}) for the vectors \vec{A} = 2\hat{i} + 3\hat{j} - 4\hat{k}, \vec{B} = -3\hat{i} + 4\hat{j} - 2\hat{k}, and \vec{C} = 7\hat{i} - 8\hat{j}. The solution involves using vector cross and dot products.

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