A line has direction ratios 6, −2, 3 and makes an obtuse angle with the positive direction of the x-axis. The direction cosines of the line are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle -\frac{6}{7}, \frac{2}{7}, -\frac{3}{7}\)
Dividing the direction ratios by \(\sqrt{36+4+9} = 7\), the direction cosines are
\( \displaystyle \pm\left(\frac{6}{7}, -\frac{2}{7}, \frac{3}{7}\right) \)
The two signs give the two opposite directions of the line. An obtuse angle with the x-axis means \(l = \cos\alpha \lt 0\), so the minus sign is taken for all three:
\( \displaystyle l = -\frac{6}{7},\quad m = \frac{2}{7},\quad n = -\frac{3}{7} \)
Why the other options are incorrect
Option (2) \(\displaystyle \frac{6}{7}, -\frac{2}{7}, \frac{3}{7}\) — Incorrect. These are direction cosines of the line, but \(l \gt 0\) gives an acute angle with the x-axis.
Option (3) \(\displaystyle -\frac{6}{7}, -\frac{2}{7}, -\frac{3}{7}\) — Incorrect. Only the signs of l and n were changed; these are not proportional to 6, −2, 3.
Option (4) \(\displaystyle -\frac{6}{7}, \frac{2}{7}, \frac{3}{7}\) — Incorrect. Only the signs of l and m were changed; these are not proportional to 6, −2, 3.