LinAlg4.2.19

Find the angle between the planes $\bar{r}\cdot(2\bar{i}-\bar{j}+\bar{k})=6,\bar{r}\cdot(\bar{i}+\bar{j}+2\bar{k})=7\,$

Given vector equations of the planes are $\bar{r}\cdot(2\bar{i}-\bar{j}+\bar{k})=6,\bar{r}\cdot(\bar{i}+\bar{j}+2\bar{k})=7\,$

where first one is m1 and the second one is m2.

Let $\theta\,$ be the angle between the given planes.

Therefore, $\cos\theta=\frac{m_1\cdot m_2}{|m_1||m_2|}\,$

$\cos\theta=\frac{(2\bar{i}-\bar{j}+\bar{k})\cdot(\bar{i}+\bar{j}+2\bar{k})}{\sqrt{4+1+1}\cdot\sqrt{1+1+4}}\,$

Hence $\cos\theta=\frac{2-1+2}{\sqrt{6}\sqrt{6}}=\frac{3}{6}=\frac{1}{2}=\cos 60\,$

Therefore $\theta=60^\circ\,$