Question

Find the angle between plane $\overrightarrow{r}.\left(\hat{i}-\hat{j}+3\hat{k}\right)=5$ and the line $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(\hat{i}-\hat{j}+\hat{k}\right)$

Answer 1

Plane: $r.(\hat{i}-\hat{j}+3\hat{k})=5$

$\overrightarrow{n}=\left(n_1,n_2,n_3\right)=(1,-1,3)$

Line: $r=(\hat{i}+\hat{j}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$

$r=\overrightarrow{a}+\lambda \overrightarrow{b}$

$\overrightarrow{a}=\left(a_1,a_2,a_3\right)=(1,1,-1)$

$\overrightarrow{b}=\left(b_1,b_2,b_3\right)=(1,-1,1)$

$\sin \theta =\left|\frac{\overrightarrow{b}.\overrightarrow{n}}{\left|\overrightarrow{b}\right|\left|\overrightarrow{n}\right|}\right|$

$\theta ={\sin }^{-1}\left|\frac{n_1{b}_1+n_2{b}_2+n_3{b}_3}{\sqrt{{n_1}^2+{n_2}^2+{n_3}^2}\sqrt{{b_1}^2+{b_2}^2+{b_3}^2}}\right|$

$={\sin }^{-1}\left|\frac{1+1+3}{\sqrt{1+1+9}\sqrt{1+1+1}}\right|$

$={\sin }^{-1}\frac{5}{\sqrt{11}\sqrt{3}}$

$={\sin }^{-1}\left(\frac{5}{\sqrt{33}}\right)$