Math Problem Statement
Find a × b and b × a if a = i + 7k and b = j.
Solution
The problem asks to compute and where:
- ,
- .
1. Cross Product Formula:
For two vectors and , their cross product is:
\mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}.$$ ### 2. Compute $$\mathbf{a} \times \mathbf{b}$$: Write the components of $$\mathbf{a}$$ and $$\mathbf{b}$$: - $$\mathbf{a} = (1, 0, 7)$$, - $$\mathbf{b} = (0, 1, 0)$$. The determinant becomes: $$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 7 \\ 0 & 1 & 0 \end{vmatrix}.$$ Expanding the determinant: $$\mathbf{a} \times \mathbf{b} = \mathbf{i} \begin{vmatrix} 0 & 7 \\ 1 & 0 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 1 & 7 \\ 0 & 0 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix}.$$ $$\mathbf{a} \times \mathbf{b} = \mathbf{i} (0 - 7) - \mathbf{j} (0 - 0) + \mathbf{k} (1 - 0).$$ $$\mathbf{a} \times \mathbf{b} = -7\mathbf{i} + 0\mathbf{j} + \mathbf{k}.$$ $$\mathbf{a} \times \mathbf{b} = -7\mathbf{i} + \mathbf{k}.$$ ### 3. Compute $$\mathbf{b} \times \mathbf{a}$$: The cross product $$\mathbf{b} \times \mathbf{a}$$ is the negative of $$\mathbf{a} \times \mathbf{b}$$: $$\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b}).$$ $$\mathbf{b} \times \mathbf{a} = -(-7\mathbf{i} + \mathbf{k}).$$ $$\mathbf{b} \times \mathbf{a} = 7\mathbf{i} - \mathbf{k}.$$ ### Final Answers: $$\mathbf{a} \times \mathbf{b} = -7\mathbf{i} + \mathbf{k},$$ $$\mathbf{b} \times \mathbf{a} = 7\mathbf{i} - \mathbf{k}.$$ --- Would you like more explanation or details? ### Related Questions: 1. What is the physical significance of the cross product in physics and mathematics? 2. How does the order of vectors affect the cross product? 3. Can the cross product be applied to non-3D vectors? 4. How is the magnitude of the cross product calculated? 5. Why is the cross product result always orthogonal to the original vectors? ### Tip: To verify the direction of the cross product, use the **right-hand rule**: Point your index finger in the direction of $$\mathbf{a}$$, your middle finger in the direction of $$\mathbf{b}$$, and your thumb will point in the direction of $$\mathbf{a} \times \mathbf{b}$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Vectors
Cross Product
Determinants
Formulas
Cross product formula: u × v = determinant |i j k|
Components of cross product: u × v = (u2v3 - u3v2)i - (u1v3 - u3v1)j + (u1v2 - u2v1)k
Theorems
Cross product theorem: u × v = -(v × u)
Suitable Grade Level
Grades 11-12