Line and Plane — Maths STD 12 Science — Question

Let θ be the acute angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Then,
$\cos \theta=\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}\right|$
$\cos \theta=\left|\frac{4(3)+(-3)(4)+5(5)}{\sqrt{4^2+(3)^2+5^2} \cdot \sqrt{3^2+4^2+5^2}}\right|$
$=\left|\frac{12-12+25}{\sqrt{16+9+25} \cdot \sqrt{9+16+25}}\right|$
$=\left|\frac{25}{50}\right|=\frac{1}{2}$
$\cos \theta=\frac{1}{2}$
$\theta=\cos ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}$
The angle between the lines is $\frac{\pi}{3}$