Question

Find the measure of the angle between the two lines whose direction ratios are $(2,2,1)$ & $(-4,1,8)$.

Answer 1

Consider

$a_1=2,b_1=2,c_1=1$

$a_2=-4,b_2=1,c_{2}8$

And,

Let $\left(\theta \right)$ be the angle between these two lines.

then

$\begin{array}{c}\cos \theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|\\ \\ \cos \theta =\left|\frac{-8+2+8}{\sqrt{4+4+1}\sqrt{16+1+64}}\right|\\ \\ \cos \theta =\frac{2}{\sqrt{9}\sqrt{81}}\\ \\ \theta ={\cos }^{-1}\left(\frac{2}{27}\right)\end{array}$