Question

Lines $\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-2}{3}$ and $\frac{x-1}{2}=\frac{y-2}{2}=\frac{z-3}{-2}$
(Check whether the lines are parallel, mutually perpendicular or intersecting in acute angle)

Answer 1

Consider the

$a_1=2,b_1=2,c_1=3.a_2=2,b_2=2,c_2=-2$

Let angle between the lines be $\theta$

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

$\cos \theta =|\frac{4+4-6}{\sqrt{4+4+9}\sqrt{4+4+4}}|$

$\cos \theta =\frac{1}{\sqrt{51}}$

$\theta ={\cos }^{-1}\left(\frac{1}{\sqrt{51}}\right)$

So, thses lines are intersecting in acute angle $\cos \left(\frac{1}{\sqrt{51}}\right)$.