1.(i) Find the angle between the planes $\mathrm{x}+2\mathrm{y}+2\mathrm{z}-5=0\text{ and }3\mathrm{x}+3\mathrm{y}+2\mathrm{z}-8=0$.
(ii) Find the angle between the planes $2\mathrm{x}-\mathrm{y}+\mathrm{z}-6=0\text{ and }\mathrm{x}+\mathrm{y}+2\mathrm{z}-7=0$

(i) Given planes are x + 2y + 2z – 5 = 0 ….(1)
$\begin{array}{l}{\mathrm{a}}_1=1,{\mathrm{b}}_1=2,{\mathrm{c}}_1=2\\ and 3\mathrm{x}+3\mathrm{y}+2\mathrm{z}-8=0 ...\left(2\right)\\ {\mathrm{a}}_2=3,{\mathrm{b}}_2=3,{\mathrm{c}}_2=2\end{array}$
Let $\mathrm{\theta }$ is the angle between  the planes (1) & (2) then
$\begin{array}{l}\cos \mathrm{\theta }=\frac{\left|{\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2+{\mathrm{c}}_1{\mathrm{c}}_2\right|}{\sqrt{{\mathrm{a}}_1^2+{\mathrm{b}}_1^2+{\mathrm{c}}_1^2}\sqrt{{\mathrm{a}}_2^2+{\mathrm{b}}_2^2+{\mathrm{c}}_2^2}}\\ \cos \mathrm{\theta }=\frac{\left|1\right(3)+2(3)+2(2\left)\right|}{\sqrt{1^2+2^2+2^2}\sqrt{3^2+3^2+2^2}}\\ =\frac{|3+6+4|}{\sqrt{9}\sqrt{22}}\\ =\frac{13}{3\sqrt{22}}\\ \Rightarrow \mathrm{\theta }={\cos }^{-1}\left(\frac{13}{3\sqrt{22}}\right)\end{array}$
(ii) Given planes are
$\begin{array}{l}2\mathrm{x}-\mathrm{y}+\mathrm{z}-6=0\to \left(1\right)\\ {\mathrm{a}}_1=2,{\mathrm{b}}_1=-1,{\mathrm{c}}_1=1\\ \mathrm{x}+\mathrm{y}+2\mathrm{z}-7=0\to \left(2\right)\\ {\mathrm{a}}_2=1,{\mathrm{b}}_2=1,{\mathrm{c}}_2=2\end{array}$
Let $\mathrm{\theta }$ is the angle between  the planes (1) & (2) then
$\begin{array}{l}\cos \mathrm{\theta }=\frac{\left|{\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2+{\mathrm{c}}_1{\mathrm{c}}_2\right|}{\sqrt{{\mathrm{a}}_1^2+{\mathrm{b}}_1^2+{\mathrm{c}}_1^2}\sqrt{{\mathrm{a}}_2^2+{\mathrm{b}}_2^2+{\mathrm{c}}_2^2}}\\ =\frac{|2-1+2|}{\sqrt{2^2+(-1{)}^2+1^2}\sqrt{1^2+1^2+2^2}}\\ =\frac{3}{\sqrt{6}\sqrt{6}}\\ =\frac{3}{6}=\frac{1}{2}\\ \cos \mathrm{\theta }=\frac{1}{2}\Rightarrow \mathrm{\theta }=\frac{\mathrm{\pi }}{3}\end{array}$