Question

Find the angle between two lines whose direction ratio are
(i) $2,1,2$ and $4,8,1$ (ii) $5,-12,13$ and $-3,4,5$

Answer 1

Angle between two lines whose direction ratios are:

$a_1,b_1,c_1$ and $a_2,b_2,c_2$ is

$\cos \theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}+\sqrt{a_2^2+b_2^2+c_2^2}}\right|$

$\left(1\right)$ $a_1=2,b_1=1,c_2=2$;

$a_2=4,b_2=8,c_2=1$

$\cos \theta =\frac{|8+8+2|}{\sqrt{4+1+4}\sqrt{6+64+1}}$

$\cos \theta =\frac{18}{3\times 9}$

$\cos \theta =\frac{2}{3}$

$\theta ={\cos }^{-1}\left(\frac{2}{3}\right)$

$\left(2\right)$ $a_1=5,b_2=-12$, $c_1=13$, $a_2=-3$, $b_2=4$, $c_2=5$

$\cos \theta =\frac{|-15-48+65|}{\sqrt{35+144+169}\sqrt{9+16+25}}$

$\cos \theta =\frac{|2|}{13\sqrt{2}5\sqrt{2}}$

$\cos \theta =\frac{1}{65}$

$\theta ={\cos }^{-1}\left(\frac{1}{65}\right)$.