Consider the line \(L: \displaystyle\frac{x-1}{1}=\frac{y+2}{-1}=\frac{z-3}{\sqrt{2}}\).
(A) The acute angle between L and the x-axis is \(\displaystyle\frac{\pi}{3}\)
(B) The acute angle between L and the y-axis is \(\displaystyle\frac{\pi}{4}\)
(C) The acute angle between L and the z-axis is \(\displaystyle\frac{\pi}{4}\)
(D) \(\displaystyle\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\) are direction cosines of L
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
The direction ratios of L are \(1, -1, \sqrt{2}\):
\(\sqrt{1+1+2} = 2\)
so its direction cosines are
\( \displaystyle \pm\left(\frac{1}{2}, -\frac{1}{2}, \frac{\sqrt{2}}{2}\right) \)
The acute angle with an axis has cosine equal to the size of the matching direction cosine:
\( \displaystyle \text{x-axis: } \frac{1}{2}\qquad \text{y-axis: } \frac{1}{2}\qquad \text{z-axis: } \frac{1}{\sqrt{2}} \)
So the acute angles are \(\frac{\pi}{3}\) with the x-axis, \(\frac{\pi}{3}\) with the y-axis and \(\frac{\pi}{4}\) with the z-axis, and the direction cosines are \(\pm\left(\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right)\).
Checking the statements
(A) The acute angle with the x-axis is \(\frac{\pi}{3}\) — Correct. Its cosine is \(\frac{1}{2}\).
(B) The acute angle with the y-axis is \(\frac{\pi}{4}\) — Incorrect. Its cosine is \(\frac{1}{2}\), so the angle is \(\frac{\pi}{3}\).
(C) The acute angle with the z-axis is \(\frac{\pi}{4}\) — Correct. Its cosine is \(\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}\).
(D) \(\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\) are direction cosines of L — Incorrect. The x- and y-cosines of L have opposite signs.