Math Problem Statement
Write the cross product of (1,2,0) and (1,4,2)
Solution
The cross product of two vectors A = (1, 2, 0) and B = (1, 4, 2) is given by the determinant of the following matrix:
\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 0 \\ 1 & 4 & 2 \\ \end{vmatrix}$$ This can be expanded as: $$\mathbf{A} \times \mathbf{B} = \mathbf{i} \begin{vmatrix} 2 & 0 \\ 4 & 2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 1 & 0 \\ 1 & 2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix}$$ Now, let's compute each of these determinants: 1. For $$\mathbf{i}$$: $$\begin{vmatrix} 2 & 0 \\ 4 & 2 \end{vmatrix} = (2 \cdot 2) - (0 \cdot 4) = 4$$ 2. For $$\mathbf{j}$$: $$\begin{vmatrix} 1 & 0 \\ 1 & 2 \end{vmatrix} = (1 \cdot 2) - (0 \cdot 1) = 2$$ 3. For $$\mathbf{k}$$: $$\begin{vmatrix} 1 & 2 \\ 1 & 4 \end{vmatrix} = (1 \cdot 4) - (2 \cdot 1) = 2$$ So, putting it all together: $$\mathbf{A} \times \mathbf{B} = 4\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$$ Therefore, the cross product is: $$\mathbf{A} \times \mathbf{B} = (4, -2, 2)$$ Would you like further details or clarifications? Here are 5 related questions for you to explore: 1. What is the physical significance of the cross product in physics? 2. How do you find the magnitude of the cross product? 3. Can the cross product be used in higher dimensions? 4. How does the cross product differ from the dot product? 5. What is the geometric interpretation of the cross product? **Tip**: The cross product is perpendicular to the plane formed by the two vectors, following the right-hand rule.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Vector Algebra
Cross Product
Determinants
Formulas
Cross product of two vectors A × B = det |i j k| |A1 A2 A3| |B1 B2 B3|
Determinant of a 2x2 matrix: |a b| |c d| = (ad - bc)
Theorems
Cross Product Theorem
Suitable Grade Level
Grades 10-12
Related Recommendation
Compute Cross Product of Vectors using Determinants: Example and Solution
Cross Product of Vectors a = (2, -1, 3) and b = (3, -2, 1)
Compute the Cross Product of Vectors Using Determinants
Cross Product of Vectors u = (2, 2, 0) and v = (2, 2, 2√2) | Compute |u × v|
Cross Product of Vectors u = (-1, 3, 2) and v = (1, 7, 0)