Math Problem Statement
how did it become negative j when u cross multiply it
Solution
Let's clarify the calculation of the cross product step by step to address the sign confusion.
Given vectors and , the cross product is computed using the determinant of a matrix:
\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 20 & 18 & -10 \\ -10 & 8 & 15 \end{vmatrix}$$ This determinant expands as follows: $$\mathbf{RAM} \times \mathbf{RAN} = \mathbf{i} \left( 18 \cdot 15 - (-10) \cdot 8 \right) - \mathbf{j} \left( 20 \cdot 15 - (-10) \cdot (-10) \right) + \mathbf{k} \left( 20 \cdot 8 - 18 \cdot (-10) \right)$$ Let's calculate each component step by step: ### $$\mathbf{i}$$ Component: $$\mathbf{i} \left( 18 \cdot 15 - (-10) \cdot 8 \right) = \mathbf{i} \left( 270 + 80 \right) = \mathbf{i} \times 350$$ ### $$\mathbf{j}$$ Component: The $$\mathbf{j}$$ component has a negative sign in front of it by the rule of determinant expansion: $$- \mathbf{j} \left( 20 \cdot 15 - (-10) \cdot (-10) \right) = - \mathbf{j} \left( 300 - 100 \right) = - \mathbf{j} \times 200$$ ### $$\mathbf{k}$$ Component: $$\mathbf{k} \left( 20 \cdot 8 - 18 \cdot (-10) \right) = \mathbf{k} \left( 160 + 180 \right) = \mathbf{k} \times 340$$ So the resulting cross product vector is: $$\mathbf{RAM} \times \mathbf{RAN} = (350, -200, 340)$$ This explains why the $$\mathbf{j}$$ component has a negative sign: it's due to the determinant expansion rule, where the middle term (the $$\mathbf{j}$$ component) always carries a negative sign. Would you like to continue with the calculations, or do you have any more questions about the steps? Here are some related questions you might consider: 1. What is the rule for expanding the determinant of a 3x3 matrix? 2. How can the cross product be interpreted geometrically in three-dimensional space? 3. Why does the middle term in a cross product carry a negative sign? 4. How do different coordinate systems affect the calculation of a cross product? 5. What are some practical applications of the cross product in physics and engineering? **Tip:** Always be careful with the signs when expanding a determinant, especially for the middle (j) component, which carries a negative sign by convention.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vectors
Cross Product
Determinants
Formulas
Cross product formula using determinants
Theorems
Properties of determinants in vector calculations
Suitable Grade Level
Advanced High School
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