Question

A line makes angles $\alpha ,\beta ,\gamma$ with the coordinate axes. if $\alpha +\beta ={90}^o$, then $\gamma =$

• A. $0$
• B. ${90}^o$
• C. ${180}^o$
• D. ${45}^o$

Answer 1

The correct option is B ${90}^o$

If a line makes the angles $\alpha ,\beta ,\gamma$,
Then, $co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$
It is given that, $\alpha +\beta ={90}^o$
=> $\alpha =\beta -{90}^o$

=> $cos\alpha =cos({90}^o-\beta )$

=> $cos\alpha =sin\beta$
=> $co{s}^2\alpha =si{n}^2\beta =1-co{s}^2\beta$
=> $co{s}^2\alpha +co{s}^2\beta =1$
As, $co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$
=> $1+co{s}^2\gamma =1$
=> $co{s}^2\gamma =0$
hence, $\gamma =\frac{\pi }{2}={90}^o$