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Direction Cosines

If α ,β ,γ ...

Question

If $α, β, γ$ are the angles which a directed line makes with the positive directions of the coordinate axes, then ${sin}^{2} α + {sin}^{2} β + {sin}^{2} γ$ is equal to

1

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4

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3

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2

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Solution

The correct option is D $2$
The direction cosines of the line are
$l^{2} + m^{2} + n^{2} = 1$
Now, $\Rightarrow {cos}^{2} α + {cos}^{2} β + {cos}^{2} γ = 1$
$\Rightarrow 1 - {sin}^{2} α + 1 - {sin}^{2} β + 1 - {sin}^{2} γ = 1$
$\Rightarrow {sin}^{2} α + {sin}^{2} β + {sin}^{2} γ = 2$

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