A vector of magnitude \(5\sqrt{3}\) makes equal acute angles with the positive directions of the coordinate axes. The vector is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(5\hat{\imath}+5\hat{\jmath}+5\hat{k}\)
Equal angles with the axes mean equal direction cosines, l = m = n. Since \(l^2+m^2+n^2 = 1\):
\(3l^2 = 1\)
\( \displaystyle l = \pm\frac{1}{\sqrt{3}} \)
The angles are acute, so the cosines are positive:
\( \displaystyle l = m = n = \frac{1}{\sqrt{3}} \)
A vector of magnitude r with direction cosines l, m, n is \(r(l\hat{\imath}+m\hat{\jmath}+n\hat{k})\):
\( \displaystyle 5\sqrt{3}\times\frac{1}{\sqrt{3}}\left(\hat{\imath}+\hat{\jmath}+\hat{k}\right) \)
\( = 5\hat{\imath}+5\hat{\jmath}+5\hat{k}\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{\sqrt{3}}\left(\hat{\imath}+\hat{\jmath}+\hat{k}\right)\) — Incorrect. This is the unit vector in that direction; its magnitude is 1.
Option (2) \(5\sqrt{3}\left(\hat{\imath}+\hat{\jmath}+\hat{k}\right)\) — Incorrect. Its magnitude is \(5\sqrt{3}\times\sqrt{3} = 15\).
Option (4) \(-5\hat{\imath}-5\hat{\jmath}-5\hat{k}\) — Incorrect. Its direction cosines are negative, so the angles with the axes are obtuse.