Question

If \( \vec{a} \) is a non-zero vector such that its projections on the vectors \( 2\hat{i} - \hat{j} + 2\hat{k},\ \hat{i} + 2\hat{j} - 2\hat{k} \), and \( \hat{k} \) are equal, then a unit vector along \( \vec{a} \) is:

• A. \( \frac{1}{\sqrt{155}} (7\hat{i} + 9\hat{j} + 5\hat{k}) \)
• B. \( \frac{1}{\sqrt{155}} (7\hat{i} + 9\hat{j} - 5\hat{k}) \)
• C. \( \frac{1}{\sqrt{155}} (7\hat{i} + 9\hat{j} + 5\hat{k}) \)
• D. \( \frac{1}{\sqrt{155}} (7\hat{i} + 9\hat{j} - 5\hat{k}) \)

Hint:

Use projection formulas and form simultaneous equations. Normalize the result to get a unit vector.

Answer 1

The Correct Option is C

Let \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) Projection of \( \vec{a} \) on a vector \( \vec{b} \) is \( \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \) Equating projections: \[ \frac{\vec{a} \cdot (2, -1, 2)}{\sqrt{9}} = \frac{\vec{a} \cdot (1, 2, -2)}{\sqrt{9}} = \frac{a_3}{1} \Rightarrow \frac{2a_1 - a_2 + 2a_3}{3} = \frac{a_1 + 2a_2 - 2a_3}{3} = a_3 \] Solving the system yields: \[ a_1 = \frac{7}{\sqrt{155}},\quad a_2 = \frac{9}{\sqrt{155}},\quad a_3 = \frac{5}{\sqrt{155}} \]