If a line makes angles \(\alpha, \beta, \gamma\) with the positive directions of x-axis, y-axis and z-axis respectively, then \(\sin^2\alpha + \sin^2\beta + \sin^2\gamma\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 2
The direction cosines of the line are \(\cos\alpha, \cos\beta, \cos\gamma\), and they satisfy
\(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\)
Using \(\sin^2\theta = 1 - \cos^2\theta\) for each angle:
\(\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma)\)
\(= 3 - 1 = 2\)
NCERT Exemplar:
Three Dimensional Geometry
