Find the angle between the following pairs of lines:
(i) $\vec r = 2\hat i - 5\hat j + \hat k + \lambda (3\hat i + 2\hat j + 6\hat k)$ and
$\vec r = 7\hat i - 6\hat k + \mu (\hat i + 2\hat j + 2\hat k)$

(ii) $\vec r = 3\hat i + \hat j - 2\hat k + \lambda (\hat i - \hat j - 2\hat k)$ and
$\vec r = 2\hat i - \hat j - 56\hat k + \mu (3\hat i - 5\hat j - 4\hat k)$

. : (i) Here, ${\vec b_1} = 3\hat i + 2\hat j + 6\hat k$ and ${\vec b_2} = \hat i + 2\hat j + 2\hat k$

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

$\cos \theta = \left| {\cfrac{{{b_1} \cdot {b_2}}}{{|{{\vec b}_1}||{{\vec b}_2}|}}} \right| = \cfrac{{|3 \times 1 + 2 \times 2 + 6 \times 2|}}{{\sqrt {{3^2} + {2^2} + {6^2}} \sqrt {{1^2} + {2^2} + {2^2}} }} = \cfrac{{19}}{{7 \times 3}} = \cfrac{{19}}{{21}}$

$\Rightarrow$ $\theta = {\cos ^{ - 1}}\left( {\cfrac{{19}}{{21}}} \right)$

(ii) Here ${\vec b_1} = \hat i - \hat j - 2\hat k$ and ${\vec b_2} = 3\hat i - 5\hat j - 4\hat k$

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

$\cos \theta = \left| {\cfrac{{{b_1} \cdot {b_2}}}{{|{{\vec b}_1}||{{\vec b}_2}|}}} \right|$

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

$= \cfrac{{16}}{{\sqrt 6 \sqrt {50} }} = \cfrac{{16}}{{\sqrt {300} }} = \cfrac{{16}}{{10\sqrt 3 }}$

$= \cfrac{8}{{5\sqrt 3 }} \Rightarrow \theta = {\cos ^{ - 1}}\left( {\cfrac{8}{{5\sqrt 3 }}} \right)$