Let $$\theta $$ be the angle between the two lines

Here direction cosines of $${x \over 2}$$ = $${y \over 2}$$ = $${z \over 1}$$ are 2, 2, 1

Also second line can be written as :

$${{x - 5} \over 2} = {{y - 2} \over {{P \over 7}}} = {{z - 3} \over 4}$$

$$ \therefore $$  its direction cosines are 2, $${{P \over 7}}$$, 4

Also, cos$$\theta $$ = $${2 \over 3}$$ (Given)

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

$$ \Rightarrow $$   $${2 \over 3}$$ $$ = \left| {{{\left( {2 \times 2} \right) + \left( {2 \times {P \over 7}} \right) + \left( {1 \times 4} \right)} \over {\sqrt {{2^2} + {2^2} + {1^2}} \sqrt {{2^2} + {{{P^2}} \over {49}} + {4^2}} }}} \right|$$

           $$ = {{4 + {{2P} \over 7} + 4} \over {3 \times \sqrt {{2^2} + {{{P^2}} \over {49}} + {4^2}} }}$$

$$ \Rightarrow $$ $${\left( {4 + {P \over 7}} \right)^2} = 20 + {{{P^2}} \over {49}}$$

$$ \Rightarrow $$  16 + $${{8P} \over 7} + {{{P^2}} \over {49}}$$ = 20 + $${{{P^2}} \over {49}}$$

$$ \Rightarrow $$ $${{8P} \over 7} = 4$$ $$ \Rightarrow $$ $$P = {7 \over 2}$$