# Find the angle between the pairs of lines: $\overrightarrow r = 2\hat i -5 \hat j +\hat k + \lambda (3\hat i +2 \hat j +6\hat k)\: and \: \overrightarrow r = 7\hat i - 6\hat k + \mu (\hat i +2\hat j +2\hat k)$

Given equations of the lines are
$\overrightarrow r = 2\hat i -5 \hat j +\hat k + \lambda (3\hat i +2 \hat j +6\hat k)$.........(i) and
$\overrightarrow r = 7\hat i - 6\hat k + \mu (\hat i +2\hat j +2\hat k)$.............(ii)
Comparing the equations with standard vector form of equation of line
$\overrightarrow r=\overrightarrow a+\lambda\:\overrightarrow b$ we get
$\overrightarrow b_1=3\hat i+2\hat j+6\hat k$ and $\overrightarrow b_2=\hat i+2\hat j+2\hat k$
$\overrightarrow b_1.\overrightarrow b_2=3+4+12=19$
$|\overrightarrow b_1|=\sqrt {9+4+36}=7$ and $|\overrightarrow b_2|=\sqrt {1+4+4}=3$
Angle between the lines (i) and (ii) is given by
$cos^{-1}\bigg(\large\frac{\overrightarrow b_1.\overrightarrow b_2}{|\overrightarrow b_1|.|\overrightarrow b_2|}\bigg)$
$=cos^{-1}\bigg(\large\frac{19}{21}\bigg)$