Question

Let \( \mathbf{a} = 3\hat{i} - \hat{j} + 2\hat{k} \), \( \mathbf{b} = \mathbf{a} \times (\hat{i} - 2\hat{k}) \) and \( \mathbf{c} = \mathbf{b} \times \hat{k} \). Then the projection of \( \mathbf{c} - 2\hat{j} \) on \( \mathbf{a} \) is:

• A. \( 2\sqrt{14} \)
• B. \( 2\sqrt{7} \)
• C. \( 3\sqrt{7} \)
• D. \( \sqrt{14} \)

Hint:

To find the projection of a vector on another, use the formula \( \text{Proj}_{\mathbf{a}} \mathbf{v} = \frac{\mathbf{a} \cdot \mathbf{v}}{|\mathbf{a}|} \) and remember to compute cross products when needed.

Answer 1

The Correct Option is A

To find the projection of \( \mathbf{c} - 2\hat{j} \) on \( \mathbf{a} \), first compute the vectors \( \mathbf{b} \) and \( \mathbf{c} \) using the given cross products. Then, use the projection formula: \[ \text{Proj}_{\mathbf{a}} \mathbf{v} = \frac{\mathbf{a} \cdot \mathbf{v}}{|\mathbf{a}|}. \] Substitute \( \mathbf{c} - 2\hat{j} \) and \( \mathbf{a} \) into the formula. 
Final Answer: \( 2\sqrt{14} \).