Which of the following statements are correct?
(A) The direction cosines of the y-axis are 0, 1, 0
(B) \(\vec{r} = \lambda\hat{k}\) is a vector equation of the z-axis
(C) The line \(\displaystyle\frac{x}{1}=\frac{y}{1}=\frac{z}{1}\) makes an angle of \(\displaystyle\frac{\pi}{4}\) with each coordinate axis
(D) If a line is perpendicular to the x-axis, then its first direction cosine is 1
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (B) only
The positive y-axis makes angles \(\frac{\pi}{2}, 0, \frac{\pi}{2}\) with the x-, y- and z-axes, so its direction cosines are
\( \displaystyle \cos\frac{\pi}{2}, \cos 0, \cos\frac{\pi}{2} = 0, 1, 0 \)
The z-axis passes through the origin with direction \(\hat{k}\), so \(\vec{r} = \vec{0}+\lambda\hat{k}\).
The line \(\frac{x}{1}=\frac{y}{1}=\frac{z}{1}\) has direction ratios 1, 1, 1:
\( \displaystyle \cos\alpha = \frac{1}{\sqrt{3}} \)
and \(\cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}\), which is different. A line perpendicular to the x-axis makes angle \(\frac{\pi}{2}\) with it:
\( \displaystyle l = \cos\frac{\pi}{2} = 0 \)
So the y-axis has direction cosines 0, 1, 0, the z-axis is \(\vec{r} = \lambda\hat{k}\), the line in (C) makes \(\cos^{-1}\frac{1}{\sqrt{3}}\) with each axis, and in (D) l = 0.
Checking the statements
(A) The direction cosines of the y-axis are 0, 1, 0 — Correct. They are \(\cos\frac{\pi}{2}, \cos 0, \cos\frac{\pi}{2}\).
(B) \(\vec{r} = \lambda\hat{k}\) is a vector equation of the z-axis — Correct. It passes through the origin along \(\hat{k}\).
(C) The line makes \(\frac{\pi}{4}\) with each axis — Incorrect. \(\cos\alpha = \frac{1}{\sqrt{3}}\), not \(\frac{1}{\sqrt{2}}\).
(D) A line perpendicular to the x-axis has first direction cosine 1 — Incorrect. It is \(\cos\frac{\pi}{2} = 0\).