If a line makes angles $α, β, γ$ with positive directions of X,Y,Z-axes, what is the value of ${s i n}^{2} α + {s i n}^{2} β + {s i n}^{2} γ$ ?

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Solution

Given , A line make a angle $α, β, χ,$ with x , y , z axis , then

let length of line be $| r |$ x , y , z be the intercept mode by line on the three coordinates axis

So $cos α = \frac{x}{r} cos β = \frac{y}{r} cos χ = \frac{z}{r}$

${cos}^{2} α + {cos}^{2} β + {cos}^{2} χ = \frac{r^{2}}{r^{2}}$

${cos}^{2} α + {cos}^{2} β + {cos}^{2} χ = 1$

$1 - {sin}^{2} α + 1 - {sin}^{2} β + 1 - {sin}^{2} χ = 1$

$∴ {sin}^{2} α + {sin}^{2} β + {sin}^{2} χ = 2$

so answer is 2

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