Question

Find the angle between the lines
$\frac{x+2}{2}=\frac{y-1}{3}=\frac{z+3}{2}$ and $\frac{x-5}{1}=\frac{y-7}{2}=\frac{z-4}{-1}$.

Answer 1

Direction ratios of line $1$, $\overrightarrow{r_1}=(2,3,2)$

Direction ratios of line $2$, $\overrightarrow{r_2}=(1,2,-1)$

Angle between lines, $\theta ={\cos }^{-1}\left(\frac{\overrightarrow{r_1}\cdot \overrightarrow{r_2}}{|\overrightarrow{r_1}||\overrightarrow{r_2}|}\right)$

$={\cos }^{-1}\left(\frac{2+6-2}{\sqrt{17}\cdot \sqrt{6}}\right)$

$={\cos }^{-1}\left(\frac{6}{\sqrt{6}\cdot \sqrt{17}}\right)$

$={\cos }^{-1}\left(\sqrt{\frac{6}{17}}\right)$.