Find the angle between the pair of lines $\frac{x-5}{3}=\frac{y+2}{5}=\frac{z+2}{4}$ and $\frac{x-1}{1}=\frac{y-3}{1}=\frac{z-3}{2}$.

The direction ratios (DRs) of the first line are $(a_1,b_1,c_1)=(3,5,4)$.The DRs of the second line are $(a_2,b_2,c_2)=(1,1,2)$.The angle $\theta$ between the lines is given by:
$$\cos \theta =\frac{|a_1{a}_2+b_1{b}_2+c_1{c}_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$
$$\cos \theta =\frac{|(3\times 1)+(5\times 1)+(4\times 2)|}{\sqrt{3^2+5^2+4^2}\sqrt{1^2+1^2+2^2}}$$
$$\cos \theta =\frac{|3+5+8|}{\sqrt{9+25+16}\sqrt{1+1+4}}=\frac{16}{\sqrt{50}\sqrt{6}}=\frac{16}{\sqrt{300}}$$
$$\cos \theta =\frac{16}{10\sqrt{3}}=\frac{8}{5\sqrt{3}}$$
Rationalizing the denominator:
$$\cos \theta =\frac{8\sqrt{3}}{5\times 3}=\frac{8\sqrt{3}}{15}$$
$$\theta ={\cos }^{-1}\left(\frac{8\sqrt{3}}{15}\right)$$