Question

Find the measure of the acute angle between the lines whose direction rations are $1,2,2$ and $-3,6,2$

Answer 1

For lines with direction ratios, $\overrightarrow{a}$ and $\overrightarrow{b}$, the acute angle $\theta$ between them is given by:
$\cos \theta =\frac{|\overrightarrow{a}\cdot \overrightarrow{b}|}{|\overrightarrow{a}||\overrightarrow{b}|}$
$\therefore \cos \theta =\frac{|1\cdot -3+2\cdot 6+2\cdot 2|}{\sqrt{1^2+2^2+2^2}\sqrt{(-3{)}^2+6^2+2^2}}=\frac{13}{\sqrt{9}\cdot \sqrt{49}}=\frac{13}{3\cdot 7}$
$⟹\theta ={\cos }^{-1}(\frac{13}{21})$