Math Problem Statement
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}.
Solution
The question asks to find the value of given the following vectors:
To solve this, we will follow these steps:
Step 1: Compute the cross product
The cross product of two vectors is given by the determinant of the following matrix:
\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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Vector Algebra
Cross Product
Dot Product
Formulas
Cross product: \vec{A} \times \vec{B} = |\hat{i} \hat{j} \hat{k}|
Dot product: \vec{C} \cdot \vec{D} = C_i D_i + C_j D_j + C_k D_k
Theorems
Properties of Vector Operations
Distributive property in vector dot products
Suitable Grade Level
College-level or Advanced High School