Question

If a line makes angles $\alpha ,\beta ,\gamma$ with positive directions of X,Y,Z-axes, what is the value of ${sin}^2\alpha +{sin}^2\beta +{sin}^2\gamma$?

Answer 1

Given , A line make a angle $\alpha ,\beta ,\chi ,$ with x , y , z axis , then

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

So $\cos \alpha =\frac{x}{r}\cos \beta =\frac{y}{r}\cos \chi =\frac{z}{r}$

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\chi =\frac{r^2}{r^2}$

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\chi =1$

$1-{\sin }^2\alpha +1-{\sin }^2\beta +1-{\sin }^2\chi =1$

$\therefore {\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\chi =2$

so answer is 2