Question

If two vectors are $ \vec{a} = \hat{i} + \hat{j} $ and $ \vec{b} = \hat{j} + \hat{k} $, then the value of $ \vec{a} \times \vec{b} $ will be

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

Hint:

To compute the cross product, use the determinant method with the unit vectors \( \hat{i}, \hat{j}, \hat{k} \).

Answer 1

The Correct Option is C

Step 1: Use the formula for the cross product of two vectors.
The cross product \( \vec{a} \times \vec{b} \) is calculated as: \[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} 1 & 1 & 0 0 & 1 & 1 \end{vmatrix} \]
Step 2: Expand the determinant.
\\ Expanding this determinant gives: \[ \vec{a} \times \vec{b} = \hat{i} \begin{vmatrix} 1 & 0 1 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 0 \\ 0 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 1 \\ 0 & 1 \end{vmatrix} \] \[ = \hat{i} (1 \times 1 - 0 \times 1) - \hat{j} (1 \times 1 - 0 \times 0) + \hat{k} (1 \times 1 - 0 \times 1) \] \[ = \hat{i} - \hat{j} + \hat{k} \]
Step 3: Conclusion.
Thus, \( \vec{a} \times \vec{b} = \hat{i} - \hat{j} + \hat{k} \).
Conclusion:
The correct answer is (C) \( \hat{i} - \hat{j} + \hat{k} \).