https://file.aiquickdraw.com/mathbot/file/17254156523539d2rdchk.jpg

The image presents a vector cross product problem, labeled as "Contoh 7" or "Example 7" in the title, which involves calculating the cross product of vectors. Specifically, it asks for the value of \(\vec{A} \times (\vec{B} \times \vec{C})\), given the three vectors:

\[
\vec{A} = \begin{pmatrix} -2.00 \\ 3.00 \\ -1.00 \end{pmatrix}
\quad
\vec{B} = \begin{pmatrix} -3.00 \\ 4.00 \\ -2.00 \end{pmatrix}
\quad
\vec{C} = \begin{pmatrix} 2.00 \\ -3.00 \\ 8.00 \end{pmatrix}
\]

To solve this, we can use the vector triple product identity, which states:

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

Let's calculate this step by step.

### Step 1: Compute Dot Products
1. \(\vec{A} \cdot \vec{C} = (-2)(2) + (3)(-3) + (-1)(8)\)
2. \(\vec{A} \cdot \vec{B} = (-2)(-3) + (3)(4) + (-1)(-2)\)

### Step 2: Compute the Final Expression
After finding the dot products, use the result to calculate the expression:

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

I'll compute these values for you now.The result of the vector triple product \(\vec{A} \times (\vec{B} \times \vec{C})\) is:

\[
\begin{pmatrix} 23.00 \\ -24.00 \\ -118.00 \end{pmatrix}
\]

If you have any further questions or would like more details on any step of the calculation, feel free to ask!

### Related Questions:
1. How is the cross product of two vectors calculated directly?
2. What is the physical interpretation of a vector triple product?
3. Can the vector triple product identity be proven geometrically?
4. How does the distributive property work with vector products?
5. What are some real-world applications of vector triple products in physics?

### Tip:
The dot product of two vectors gives a scalar that represents how much one vector projects onto another, which is a crucial concept in many fields, including physics and engineering.

For the following three vectors, what is \(\vec{A} \times (\vec{B} \times \vec{C})\)?

\[
\vec{A} = \begin{pmatrix} -2.00 \\ 3.00 \\ -1.00 \end{pmatrix},
\vec{B} = \begin{pmatrix} -3.00 \\ 4.00 \\ -2.00 \end{pmatrix},
\vec{C} = \begin{pmatrix} 2.00 \\ -3.00 \\ 8.00 \end{pmatrix}
\]

Learn how to solve the vector triple product \(\vec{A} \times (\vec{B} \times \vec{C})\) using cross product and dot product concepts. This detailed step-by-step guide covers key mathematical principles, including the vector triple product identity and related formulas. Suitable for high school students in Grades 11-12.

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