If a unit vector \( \vec{a} \) makes equal acute angles with axes, then the projection of \( \vec{a} \) on \( \vec{b}=5\hat{\imath}+7\hat{\jmath}-\hat{k} \), is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{11}{15} \)
Equal acute angles give direction cosines \( \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} \), so
\( \displaystyle \vec{a}=\frac{1}{\sqrt{3}}(\hat{\imath}+\hat{\jmath}+\hat{k}) \)
\( \displaystyle \text{Projection}=\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|} \)
\( \displaystyle =\frac{(5+7-1)/\sqrt{3}}{\sqrt{75}} \)
\( \displaystyle =\frac{11}{\sqrt{3}\times 5\sqrt{3}} \)
\( \displaystyle =\frac{11}{15} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{11}{5\sqrt{3}} \) — Incorrect. This forgets the factor \( \frac{1}{\sqrt{3}} \) of the unit vector: \( \sqrt{3}\times 5\sqrt{3}=15 \), not \( 5\sqrt{3} \).
Option (3) \( \displaystyle\frac{4}{5} \) — Incorrect. The projection is \( \frac{11}{15} \), not \( \frac{4}{5} \).
Option (4) \( \displaystyle\frac{4}{5\sqrt{3}} \) — Incorrect. The projection is \( \frac{11}{15} \), not \( \frac{4}{5\sqrt{3}} \).