Maharashtra BoardEnglish MediumSTD 12 ScienceMathsLine and Plane2 Marks
Question
Find the angle between planes $\bar{r} \cdot(\hat{i}+\hat{j}-2 \hat{k})=8$ and $\bar{r} \cdot(-2 \hat{i}+\hat{j}+\hat{k})=3$
✓
Answer
Normal to the given planes are $\bar{n}_1=\hat{i}+\hat{j}-2 \hat{k}$ and $\bar{n}_2=-2 \hat{i}+\hat{j}+\hat{k}$ The acute angle $\theta$ between normal is given by $ \begin{aligned} & \cos \theta=\left|\frac{\bar{n}_1 \cdot \bar{n}_2}{\left|\bar{n}_1\right| \cdot\left|\bar{n}_2\right|}\right| \\ & \therefore \cos \theta=\left|\frac{\mid(\hat{i}+\hat{j}-2 \hat{k}) \cdot(-2 \hat{i}+\hat{j}+\hat{k})}{\sqrt{6} \cdot \sqrt{6}}\right|=\left|\frac{-3}{6}\right|=\frac{1}{2} \\ & \therefore \cos \theta=\frac{1}{2} \quad \therefore \theta=60^{\circ}=\frac{\pi}{3} \end{aligned} $ The acute angle between normals $\bar{n}_1$ and $\bar{n}_2$ is $60^{\circ}$. $\therefore \quad$ The angle between given planes is $60^{\circ}$.
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