Question

Find the angle between planes $\overline{r}.(\hat{i}-2\hat{j}+3\hat{k})=5$ and the line $\overline{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$

Answer 1

We know that the angle between a line $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ and a plane $\overrightarrow{r}.\overrightarrow{n}=d$ is given by:

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

therefore,

$\begin{array}{c}\sin \theta =\frac{\left(\hat{i}-2\hat{j}+3\hat{k}\right).\left(\hat{i}-\hat{j}+\hat{k}\right)}{\sqrt{1^2+2^2+3^2}.\sqrt{1^2+1^2+1^2}}=\frac{6}{\sqrt{42}}=\sqrt{\frac{6}{7}}\\ \theta ={\sin }^{-1}\left(\sqrt{\frac{6}{7}}\right)\end{array}$