Math Problem Statement
For the following three vectors, what is \(\vec{A} \times (\vec{B} \times \vec{C})\)?
\[
\vec{A} = \begin{pmatrix} -2.00 \\ 3.00 \\ -1.00 \end{pmatrix},
\vec{B} = \begin{pmatrix} -3.00 \\ 4.00 \\ -2.00 \end{pmatrix},
\vec{C} = \begin{pmatrix} 2.00 \\ -3.00 \\ 8.00 \end{pmatrix}
\]
Solution
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 , given the three vectors:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vector Cross Product
Dot Product
Vector Triple Product
Formulas
Vector triple product identity: \(\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{A} \cdot \vec{B})\vec{C}\)
Theorems
Vector Triple Product Theorem
Suitable Grade Level
Grades 11-12