Math Problem Statement
Solution
To solve the problem, we need to evaluate the expression , where the vectors are given as:
Step 1: Calculate the cross product
The cross product is computed using the determinant of a matrix composed of the unit vectors and the components of vectors and :
\hat{i} & \hat{j} & \hat{k} \\ 3.00 & 3.00 & -4.00 \\ -3.00 & 2.00 & 2.00 \end{vmatrix}$$ Expanding this determinant: $$\vec{A} \times \vec{B} = \hat{i}\left(3.00 \times 2.00 - (-4.00) \times 2.00\right) - \hat{j}\left(3.00 \times 2.00 - (-4.00) \times (-3.00)\right) + \hat{k}\left(3.00 \times 2.00 - 3.00 \times (-3.00)\right)$$ Calculating each component: $$\vec{A} \times \vec{B} = \hat{i}(6.00 + 8.00) - \hat{j}(6.00 + 12.00) + \hat{k}(6.00 + 9.00)$$ $$\vec{A} \times \vec{B} = \hat{i}(14.00) - \hat{j}(18.00) + \hat{k}(15.00)$$ $$\vec{A} \times \vec{B} = 14.00\hat{i} - 18.00\hat{j} + 15.00\hat{k}$$ ### Step 2: Compute the dot product $$\vec{C} \cdot (\vec{A} \times \vec{B})$$ Now, the dot product of $$\vec{C}$$ and $$\vec{A} \times \vec{B}$$ is: $$\vec{C} \cdot (\vec{A} \times \vec{B}) = (7.00\hat{i} - 9.00\hat{j}) \cdot (14.00\hat{i} - 18.00\hat{j} + 15.00\hat{k})$$ Only the $$\hat{i}$$ and $$\hat{j}$$ components contribute, since $$\vec{C}$$ has no $$\hat{k}$$ component: $$\vec{C} \cdot (\vec{A} \times \vec{B}) = 7.00 \times 14.00 + (-9.00) \times (-18.00)$$ $$\vec{C} \cdot (\vec{A} \times \vec{B}) = 98.00 + 162.00 = 260.00$$ ### Step 3: Calculate $$2 \cdot \vec{C} \cdot (\vec{A} \times \vec{B})$$ Finally, multiply the result by 2: $$2 \cdot 260.00 = 520.00$$ ### Final Answer: $$2 \cdot \vec{C} \cdot (\vec{A} \times \vec{B}) = 520.00$$ Would you like more details or have any questions? --- Here are 5 related questions you might find useful: 1. How do you compute the magnitude of a cross product vector? 2. What is the geometric interpretation of the dot product between two vectors? 3. How does the cross product differ from the dot product in terms of the resulting vector or scalar? 4. Can the cross product of two parallel vectors be non-zero? Why or why not? 5. How do you determine the direction of the resultant vector from a cross product? **Tip:** Remember that the cross product is anti-commutative, meaning $$\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vector Operations
Cross Product
Dot Product
Formulas
Cross Product: A × B = |i j k|
|Ax Ay Az|
|Bx By Bz|
Dot Product: A ⋅ B = Ax * Bx + Ay * By + Az * Bz
Theorems
Cross Product Resultant Vector
Dot Product Scalar Result
Suitable Grade Level
Grades 11-12
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