A line has direction ratios k, 2k, 2, where k > 0. If the direction cosine of the line with respect to the z-axis is \(\displaystyle\frac{2}{3}\), then k is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1
The direction cosines are the direction ratios divided by \(\sqrt{a^2+b^2+c^2}\):
\(\sqrt{k^2+4k^2+4} = \sqrt{5k^2+4}\)
The third direction cosine is given:
\( \displaystyle \frac{2}{\sqrt{5k^2+4}} = \frac{2}{3} \)
\(5k^2+4 = 9\)
\(k^2 = 1\)
Since k > 0, \(k = 1\). The direction cosines are then \(\frac{1}{3}, \frac{2}{3}, \frac{2}{3}\).