Question

If the vector $\overline{r}=\lambda \hat{i}+2\hat{j}-\hat{k}$ along a line makes angles $\alpha ,\beta ,\gamma$ with the coordinate axes then ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =......$

• A. $1$
• B. $2$
• C. $1+{\lambda }^2$
• D. $1-{\lambda }^2$

Answer 1

The correct option is B $2$

$\overline{r}=\lambda \stackrel{\wedge }{i}+2\stackrel{\wedge }{j}-\stackrel{\wedge }{k}$

$⟹\cos \alpha =\frac{\lambda }{\sqrt{{\lambda }^2+4+1}}=\frac{\lambda }{\sqrt{{\lambda }^2+5}}$

$\cos \beta =\frac{2}{\sqrt{{\lambda }^2+5}}=\frac{2}{\sqrt{{\lambda }^2+5}}$

$\cos \gamma =\frac{-1}{\sqrt{{\lambda }^2+5}}=\frac{-1}{\sqrt{{\lambda }^2+5}}$

$\cos {}^2\alpha +\cos {}^2\beta +\cos {}^2\gamma =\frac{{\lambda }^2+5}{{\lambda }^2+5}=1$

$⟹(1-sin{}^2\alpha )+(1-sin{}^2\beta )+(1-sin{}^2\gamma )=1$

$⟹sin{}^2\alpha +sin{}^2\beta +sin{}^2\gamma =2$

$\therefore B)Ans.$