Question

Find the angle between the lines joining the points $(1,4,2),(-2,1,2)$ and $(1,2,3),(2,5,1)$

Answer 1

Equation of the line joining $(1,4,2)(-2,1,2)$ is

$\frac{x-1}{-2-1}=\frac{y-4}{1-4}=\frac{z-2}{2-2}$

( or )

$\frac{x-1}{-3}=\frac{y-4}{-3}=\frac{z-2}{0}$

Equation of the line joining $(1,2,3)(2,5,1)$ is

$\frac{x-1}{2-1}=\frac{y-2}{5-2}=\frac{z-3}{1-3}$

(or)

$\frac{x-1}{1}=\frac{y-2}{3}=\frac{z-3}{-2}$

$\cos \alpha =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}{a}_2^2+b_2^2+c_2^2}$ $\{where{a}_1=-3b_1=-3c_1=0a_2=1,b_2=3c_2=-2$

$=\frac{-3\times 1+(-3)\times 3+0}{\sqrt{9+9+0}\sqrt{1+9+4}}$

$=\frac{-3-9}{\sqrt{18}\sqrt{14}}=\frac{-12}{3\sqrt{2}\sqrt{14}}=\frac{-4\sqrt{2}}{2\sqrt{14}}$

$=-2\sqrt{\frac{2}{14}}=\frac{-2}{\sqrt{7}}$

$\alpha ={\cos }^{-1}(-2\sqrt{7})$