Question

Find the angle between the planes $\overrightarrow{r}.\left(\hat{i}-2\hat{j}+3\hat{k}\right)=5$ 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$:\overrightarrow{r}.(\hat{i}-2\hat{j}+3\hat{k})=5$ Line :$\overrightarrow{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$

angle between line and plane is complement of angle between line and its normal .

$\cos \theta =\frac{1\times 1+-2\times -1+3\times 1}{\sqrt{(1^2+2^2+3^2)(1^2+1^2+1^2)}}=\frac{\sqrt{42}}{7}\Rightarrow \theta ={\cos }^{-1}\frac{\sqrt{42}}{7}$

$\therefore$ angle between plane and line $=90-\theta =90-{\cos }^{-1}(\frac{\sqrt{42}}{7})=67.{79}^{\circ }$