Math Problem Statement
if vector a = 2i+3j+4k and vector b = 7i+8j-6k find a x b
Solution
To find the cross product of two vectors and , you can use the following determinant formula:
\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 7 & 8 & -6 \end{vmatrix}$$ This determinant expands as: $$\mathbf{a} \times \mathbf{b} = \mathbf{i} \left( 3 \cdot (-6) - 4 \cdot 8 \right) - \mathbf{j} \left( 2 \cdot (-6) - 4 \cdot 7 \right) + \mathbf{k} \left( 2 \cdot 8 - 3 \cdot 7 \right)$$ Now, calculate each component: - The $$\mathbf{i}$$ component: $$3 \cdot (-6) - 4 \cdot 8 = -18 - 32 = -50$$ - The $$\mathbf{j}$$ component: $$2 \cdot (-6) - 4 \cdot 7 = -12 - 28 = -40$$ - The $$\mathbf{k}$$ component: $$2 \cdot 8 - 3 \cdot 7 = 16 - 21 = -5$$ Thus, the cross product $$\mathbf{a} \times \mathbf{b}$$ is: $$\mathbf{a} \times \mathbf{b} = -50\mathbf{i} + 40\mathbf{j} - 5\mathbf{k}$$ Or in vector form: $$\mathbf{a} \times \mathbf{b} = -50\mathbf{i} + 40\mathbf{j} - 5\mathbf{k}$$ Would you like more details on any part of this solution or have any questions? Here are some related questions to explore: 1. How do you find the magnitude of the cross product $$\mathbf{a} \times \mathbf{b}$$? 2. What is the physical interpretation of the cross product? 3. How do you verify that $$\mathbf{a} \times \mathbf{b}$$ is perpendicular to both $$\mathbf{a}$$ and $$\mathbf{b}$$? 4. What is the cross product of $$\mathbf{a}$$ with itself? 5. How does the cross product relate to the area of a parallelogram formed by two vectors? **Tip:** Remember that the cross product results in a vector that is perpendicular to the plane formed by the original vectors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vectors
Cross Product
Formulas
Cross product determinant formula
Theorems
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Suitable Grade Level
Advanced High School