Math Problem Statement
Solution
We are tasked with finding the cross product of vectors u and v, and then checking if the result is orthogonal to both u and v.
Given:
- u =
- v =
To compute the cross product u × v, we can use the following determinant formula:
\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$$ Substituting the values of **u** and **v**: $$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 3 & -4 \\ 3 & 0 & -3 \end{vmatrix}$$ This expands to: $$\mathbf{i} \left( 3 \times -3 - (-4) \times 0 \right) - \mathbf{j} \left( 0 \times -3 - (-4) \times 3 \right) + \mathbf{k} \left( 0 \times 0 - 3 \times 3 \right)$$ Simplifying each term: - For $$\mathbf{i}$$: $$3 \times -3 = -9$$, so the coefficient of $$\mathbf{i}$$ is $$-9$$. - For $$\mathbf{j}$$: $$-(-4 \times 3) = 12$$, so the coefficient of $$\mathbf{j}$$ is $$-12$$. - For $$\mathbf{k}$$: $$-9$$, so the coefficient of $$\mathbf{k}$$ is $$-9$$. Thus: $$\mathbf{u} \times \mathbf{v} = \langle -9, -12, -9 \rangle$$ ### Checking orthogonality: A vector is orthogonal to another if their dot product is zero. 1. **u · (u × v)**: $$\mathbf{u} \cdot \mathbf{u} \times \mathbf{v} = 0 \times -9 + 3 \times -12 + (-4) \times -9 = -36 + 36 = 0$$ 2. **v · (u × v)**: $$\mathbf{v} \cdot \mathbf{u} \times \mathbf{v} = 3 \times -9 + 0 \times -12 + (-3) \times -9 = -27 + 27 = 0$$ Since both dot products are zero, the cross product is orthogonal to both **u** and **v**. Would you like more details or further clarification? ### Related Questions: 1. What is the geometric interpretation of the cross product? 2. Can the magnitude of the cross product be calculated, and what does it represent? 3. How does the right-hand rule help in determining the direction of the cross product? 4. Can you calculate the angle between **u** and **v** from the cross product? 5. What happens when two vectors are parallel? What is their cross product? ### Tip: Always check the orthogonality by computing dot products, as it confirms the correctness of the cross product computation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vector Algebra
Cross Product
Orthogonality
Formulas
Cross product formula: u × v = |i j k| |u1 u2 u3| |v1 v2 v3|
Orthogonality check using dot product: u · (u × v) = 0 and v · (u × v) = 0
Theorems
Orthogonality Theorem
Suitable Grade Level
Grades 10-12
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