If a vector makes angles \(\alpha\), \(\beta\) and \(\gamma\) with the positive directions of the 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 \(\cos\alpha\), \(\cos\beta\), \(\cos\gamma\) of any vector satisfy
\(\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1\)
Writing each \(\sin^2\) as \(1-\cos^2\):
\(\sin^2\alpha+\sin^2\beta+\sin^2\gamma = 3-(\cos^2\alpha+\cos^2\beta+\cos^2\gamma)\)
\( = 3-1 = 2\)