Which of the following statements are correct?
(A) 1, 1, 1 can be the direction cosines of a line
(B) \(\displaystyle\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}\) can be the direction cosines of a line
(C) A line can make an angle of 45° with each of the positive coordinate axes
(D) If a line makes an angle of 90° with the positive z-axis and 45° with the positive x-axis, then it makes an angle of 45° or 135° with the positive y-axis
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (B) and (D) only
Three numbers l, m, n can be direction cosines only when \(l^2+m^2+n^2 = 1\). For angles \(\alpha, \beta, \gamma\) with the axes this reads \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1\).
(A): \(1+1+1 = 3\)
\( \displaystyle \text{(B): } \frac{1}{2}+\frac{1}{4}+\frac{1}{4} = 1 \)
\( \displaystyle \text{(C): } \cos^2 45^\circ \times 3 = \frac{3}{2} \)
For (D):
\( \displaystyle \cos^2\beta = 1-\cos^2 45^\circ-\cos^2 90^\circ = 1-\frac{1}{2}-0 = \frac{1}{2} \)
\( \displaystyle \cos\beta = \pm\frac{1}{\sqrt{2}} \)
So the sums of squares are 3 in (A), 1 in (B) and \(\frac{3}{2}\) in (C), and in (D) \(\cos\beta = \pm\frac{1}{\sqrt{2}}\).
Checking the statements
(A) 1, 1, 1 can be the direction cosines of a line — Incorrect. Their squares add to 3, not 1.
(B) \(\frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}\) can be the direction cosines of a line — Correct. Their squares add to exactly 1.
(C) A line can make 45° with each positive axis — Incorrect. The squares of the cosines add to \(\frac{3}{2}\), not 1.
(D) The angle with the positive y-axis is 45° or 135° — Correct. \(\cos\beta = \pm\frac{1}{\sqrt{2}}\) gives \(\beta = 45^\circ\) or \(135^\circ\).