Question

Find the acute angle between the lines whose direction ratios are $(5,12,-13)$ and $(3,-4,5)$.

Answer 1

$a_1=5,b_1=12,c_1=-13$
$a_2=3,b_2=-4,c_2=5$
Acute angle $\cos \theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{{a_1}^2+{b_1}^2+{c_1}^2}\sqrt{{a_2}^2+{b_2}^2+{c_2}^2}}\right|$
$\Rightarrow \cos \theta =\left|\frac{15-48-65}{\sqrt{25+144+169}\sqrt{9+16+25}}\right|$
$\Rightarrow \cos \theta =\left|\frac{-98}{13\sqrt{2}\times 5\times \sqrt{2}}\right|$
$\Rightarrow \cos \theta =\frac{-98}{13\times 5\times 2}$
$\Rightarrow \cos \theta =|-\frac{49}{65}|$
$\Rightarrow \cos \theta =\frac{49}{65}\Rightarrow \theta ={\cos }^{-1}\left(\frac{49}{65}\right)$