Question

If \( \mathbf{a} = ( \hat{i} + \hat{j} + \hat{k} ) \), \( \mathbf{a} \times \mathbf{b} = \hat{i} - \hat{j} \), then \( \mathbf{b} \) is:

• A. \( \hat{i} + 2\hat{k} \)
• B. \( 2\hat{i} + 2\hat{k} \)
• C. \( 2\hat{i} - 2\hat{k} \)
• D. \( 2\hat{i} + 2\hat{j} \)

Hint:

To solve for vectors in cross product problems, use the properties of the cross product and compare the resulting vector with the given values.

Answer 1

The Correct Option is B

Step 1: The cross product of \( \mathbf{a} \) and \( \mathbf{b} \) gives the vector perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). Use the given cross product result to determine \( \mathbf{b} \).
Step 2: After solving for \( \mathbf{b} \), we find \( \mathbf{b} = 2\hat{i} + 2\hat{k} \).

Final Answer: \[ \boxed{2\hat{i} + 2\hat{k}} \]