Question

Find the angle between planes $\overline{r}.(\hat{i}-2\hat{j}+\hat{k})=5$ and the line $\overline{r}=(\hat{i}+\hat{j}-\hat{k})-\lambda (\hat{i}-\hat{j}+\hat{k})$

Answer 1

Direction ratios of the Normal to the given plane are $a_1=1,b_1=-2,c_1=1$

and Direction ratios of the vector parallel to given line are $a_2=1,b_2=-1,c_2=1$

Let $\theta$ be the angle between the given plane and line then,

$\sin \theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{{a_1}^2+{b_1}^2+{c_1}^2}\sqrt{{a_2}^2+{b_2}^2+{c_2}^2}}\right|$

$=\left|\frac{1+2+1}{\sqrt{1+4+1}\sqrt{1+1+1}}\right|$

$=\left|\frac{4}{\sqrt{6}\sqrt{3}}\right|$

$=\frac{4}{\sqrt{18}}$

$=\frac{4}{3\sqrt{2}}$

$\Rightarrow \theta ={\sin }^{-1}\left(\frac{4}{3\sqrt{2}}\right)$