Math Problem Statement
Given u = <3, −1, 5> and v = <6, 10, 4>, find the cross product u × v
Solution
To find the cross product where and , we can use the determinant method. The formula for the cross product in three dimensions is given by:
\mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$$ Substituting the components of $$\mathbf{u}$$ and $$\mathbf{v}$$: $$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & -1 & 5 \\ 6 & 10 & 4 \end{vmatrix}$$ Calculating the determinant, we expand it as follows: $$\mathbf{u} \times \mathbf{v} = \mathbf{i} \begin{vmatrix} -1 & 5 \\ 10 & 4 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 3 & 5 \\ 6 & 4 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 3 & -1 \\ 6 & 10 \end{vmatrix}$$ Calculating each of the 2x2 determinants: 1. $$\begin{vmatrix} -1 & 5 \\ 10 & 4 \end{vmatrix} = (-1)(4) - (5)(10) = -4 - 50 = -54$$ 2. $$\begin{vmatrix} 3 & 5 \\ 6 & 4 \end{vmatrix} = (3)(4) - (5)(6) = 12 - 30 = -18$$ 3. $$\begin{vmatrix} 3 & -1 \\ 6 & 10 \end{vmatrix} = (3)(10) - (-1)(6) = 30 + 6 = 36$$ Now substituting back, we have: $$\mathbf{u} \times \mathbf{v} = -54 \mathbf{i} - (-18) \mathbf{j} + 36 \mathbf{k}$$ $$= -54 \mathbf{i} + 18 \mathbf{j} + 36 \mathbf{k}$$ Thus, the cross product $$\mathbf{u} \times \mathbf{v}$$ is: $$\mathbf{u} \times \mathbf{v} = \langle -54, 18, 36 \rangle$$ ### Summary The cross product $$\mathbf{u} \times \mathbf{v} = \langle -54, 18, 36 \rangle$$. Do you want further details or explanations? Here are some related questions: 1. What is the geometric interpretation of the cross product? 2. How does the cross product relate to the area of a parallelogram? 3. Can you calculate the magnitude of the cross product? 4. What are the properties of the cross product? 5. How can the cross product be applied in physics? **Tip:** Remember that the cross product is anti-commutative, meaning $$\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vector Calculus
Cross Product
Formulas
Cross product formula \( \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} \)
Theorems
Properties of the cross product
Anti-commutative property of cross product
Suitable Grade Level
Grades 10-12
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