Question

If the line $\overrightarrow{r}=2\hat{i}-\hat{j}+3\hat{k}+\lambda (\hat{i}+\hat{j}+\sqrt{2}\hat{k})$ makes angles $\alpha ,\beta ,\gamma$ with $yz,xz$ and $xy$ planes respectively, then $si{n}^2\alpha +si{n}^2\beta +si{n}^2\gamma$ is

• A. 3
• B. 2
• C. 1
• D. 0

Answer 1

The correct option is C

1

The vectors normal to $xy,yz$ and $zx$ are $\hat{k},\hat{i}$ and $\hat{j}$ respectively and the angle made by the line with $xy$ plane is $\alpha$

Let ${\theta }_1,{\theta }_2,{\theta }_3$ be the angles made by the line with the coordinate axes.

$\overrightarrow{V}=\hat{i}+\hat{j}+\sqrt{2}\hat{k}$

The angle made by the line with X-coordinate is ${\theta }_1$

$\therefore \cos {\theta }_1=\left|\frac{\overrightarrow{V}.\hat{k}}{|\overrightarrow{V}|\hat{k}}\right|=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}$

$\therefore$ the angle made by the line with $yz$ plane; $\alpha ={90}^{\circ }-{\theta }_1\Rightarrow \sin \alpha =\frac{1}{\sqrt{2}})$

Similarly, the angle made by the line with Y-coordinate is ${\theta }_2$

$\Rightarrow \cos {\theta }_2=\left|\frac{\overrightarrow{V}.\hat{i}}{|\overrightarrow{V}||\hat{i}|}\right|=\frac{1}{2}$

$\therefore$ the angle made by the line with $xz$ plane; $\beta ={90}^{\circ }-{\theta }_2\Rightarrow \sin \beta =\frac{1}{2})$

And the angle made by the line with Z-coordinate is ${\theta }_3$

$\Rightarrow \cos {\theta }_3=\left|\frac{\overrightarrow{V}.\hat{j}}{|\overrightarrow{V}||\hat{j}|}\right|=\frac{1}{2}$

$\therefore$ the angle made by the line with $xy$ plane; $\gamma ={90}^{\circ }-{\theta }_3\Rightarrow \sin \gamma =\frac{1}{2})$

$\Rightarrow {\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =1$