Find the angle between the following pair of lines:
(i) $\cfrac{{x - 2}}{2} = \cfrac{{y - 1}}{5} = \cfrac{{z + 3}}{{ - 3}}$ and $\cfrac{{x + 2}}{{ - 1}} = \cfrac{{y - 4}}{8} = \cfrac{{z - 5}}{4}$
(ii) $\cfrac{x}{2} = \cfrac{y}{2} = \cfrac{z}{1}$ and $\cfrac{{x - 5}}{4} = \cfrac{{y - 2}}{1} = \cfrac{{z - 3}}{8}$

(i) Direction ratios of the given lines are respectively

$< 2,5, - 3 >$ and $< - 1,8,4 >$
Let $\theta$ be the angle between the given lines,

then
$\cos \theta = \left| {\cfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|,$

where ${a_1},{b_2},{c_1}$ and ${a_2},{b_2},{c_2}$ are direction ratios

$\Rightarrow$ $\cos \theta = \cfrac{{|2 \times ( - 1) + 5 \times 8 + ( - 3) \times 4|}}{{\sqrt {{2^2} + {5^2} + {{( - 3)}^2}} \sqrt {{{( - 1)}^2} + {8^2} + {4^2}} }}$

$= \cfrac{{26}}{{\sqrt {38} \sqrt {81} }} = \left( {\cfrac{{26}}{{9\sqrt {38} }}} \right)$

$\Rightarrow$ $\theta = {\cos ^{ - 1}}\left( {\cfrac{{26}}{{9\sqrt {38} }}} \right)$

(ii) Direction ratios of the given lines are respectively $< 2,2,1 >$ and $< 4,1,8 >$

Let $\theta$ be the angle between the given lines,

then
$\cos \theta = \left| {\cfrac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|$

$\Rightarrow$ $\cos \theta = \cfrac{{|2 \times 4 + 2 \times 1 + 1 \times 8|}}{{\sqrt {{2^2} + {2^2} + {1^2}} \sqrt {{4^2} + {1^2} + {8^2}} }}$

$= \cfrac{{18}}{{\sqrt 9 \sqrt {81} }} = \cfrac{{18}}{{3 \times 9}} = \cfrac{2}{3}$