Question

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

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

Answer 1

The correct option is A ${90}^{\circ }$

Given : $\alpha +\beta ={90}^{\circ }$
$\Rightarrow {\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1\cdots \left(i\right)$
$\Rightarrow \alpha ={90}^{\circ }-\beta$
$\Rightarrow \cos \alpha =\sin \beta$
$\Rightarrow {\cos }^2\alpha ={\sin }^2\beta \Rightarrow {\cos }^2\alpha +{\cos }^2\beta =1$
Using $\left(i\right),$ we get
${\cos }^2\gamma =0$
$\Rightarrow \gamma ={90}^{\circ }$