A vector \(\vec a\) of magnitude \(3\sqrt{2}\) making an angle of \(\displaystyle\frac{\pi}{3}\) with \(\hat{\imath}\), \(\displaystyle\frac{\pi}{4}\) with \(\hat{\jmath}\) and an acute angle \(\theta\) with \(\hat{k}\), is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle 3\sqrt{2}\left(\frac{1}{2}\hat{\imath},\frac{1}{\sqrt{2}}\hat{\jmath},\frac{1}{2}\hat{k}\right)\)
A vector of magnitude r with direction cosines l, m, n is \(r(l\hat{\imath} + m\hat{\jmath} + n\hat{k})\), and \(l^2 + m^2 + n^2 = 1\).
\(\displaystyle l = \cos\frac{\pi}{3} = \frac{1}{2},\quad m = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}\)
\(\displaystyle n^2 = 1 - \frac{1}{4} - \frac{1}{2} = \frac{1}{4}\)
Since \(\theta\) is acute, \(n = \cos\theta \gt 0\):
\(\displaystyle n = \frac{1}{2}\)
So
\(\displaystyle \vec a = 3\sqrt{2}\left(\frac{1}{2}\hat{\imath} + \frac{1}{\sqrt{2}}\hat{\jmath} + \frac{1}{2}\hat{k}\right)\)
This is Option (2), which lists the three components separated by commas.
Shortcut: Only Option (2) has \(\frac{1}{2}\) with \(\hat{\imath}\) and \(\frac{1}{\sqrt{2}}\) with \(\hat{\jmath}\). In Option (1) the squares add up to \(\frac{1}{2} + \frac{1}{2} + \frac{1}{4} = \frac{5}{4}\), and in Option (4) to \(\frac{3}{4}\), so these are not direction cosines at all.