Question

Find the angle between the line $\frac{x-1}{1}=\frac{y-2}{-1}=\frac{z+1}{1}$ and the plane 2x + y − z = 4.

Answer 1

$$\begin{gathered} \text{The given line is parallel to the vector}\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\text{and the given plane is normal to the vector}\overrightarrow{n}=2\hat{i}+\hat{j}-\hat{k}. \\ \text{We know that the angle θ between the line and the plane is given by} \\ \sin \theta =\frac{\overrightarrow{b}.\overrightarrow{n}}{\left|\overrightarrow{b}\right|\left|\overrightarrow{n}\right|} \\ =\frac{\left(\hat{i}-\hat{j}+\hat{k}\right).\left(2\hat{i}+\hat{j}-\hat{k}\right)}{\left|\hat{i}-\hat{j}+\hat{k}\right|\left|2\hat{i}+\hat{j}-\hat{k}\right|}=\frac{2-1-1}{\sqrt{1+1+1}\sqrt{4+1+1}}=0 \\ \Rightarrow \theta ={\sin }^{-1}\left(0\right)=0 \end{gathered}$$