If $α, β, γ$ are the angles made by a vector with the coordinate axes in the positive direction, then the range of $sin α sin β + sin β sin γ + sin γ sin α$ is

[- 1, 1]

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[- \frac{1}{2}, 1]

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[- 1, 2]

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[0, 1]

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Solution

The correct option is C $[- 1, 2]$
When $α, β$ and $γ$ are the angles made with co-ordinate axes, we know that
${cos}^{2} α$ + ${cos}^{2} β + {cos}^{2} γ = 1$
hence ${sin}^{2} α + {sin}^{2} β + {sin}^{2} γ = 2$
Now taking two extreme case, one that $sin α = sin β = sin γ = \frac{2}{3}$ and the second one, $sin α = sin β = - 1 sin γ = 0.$
Thus the value of $sin α sin β + sin α sin γ + sin β sin γ$ becomes $2$ for the first case and becomes $- 1$ for the second case and thus we get the answer.

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