Question

Find the acute angle between the lines passing through (-3 , -1 , 0), (2, -3 , 1) and (1 , 2 , 3),(-1 ,4 , -2) respectively.

Answer 1

Let the given points are

$\begin{array}{c}A\left(-3,-1,0\right)\\ B\left(2.-3.1\right)\\ C\left(1,2,3\right)\\ D\left(-1,4,-2\right)\end{array}$

Direction ratios of line $AB$ are

$2+3,-3+1,1-0$

$5,-2,1$

Let $a_1=5,b_1=-2,c_1=1$

Direction ratio opf the line $CD$ are

$\begin{array}{c}{a}_2=-1-1=-2\\ b_2=4-2=2\\ c_2=-2-3=-5\end{array}$

let $O$ be the acute angle between line $CD$ and$AB$

then

$\begin{array}{c}\cos \theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{{a^2}_1+{b_1}^2+{c_1}^2}\sqrt{{a_2}^2+{b_2}^2+{c_2}^2}}\right|\\ \cos \theta =\left|\frac{-10-4-5}{\sqrt{25+4+1}\sqrt{4+4+25}}\right|\\ \cos \theta =\frac{-19}{\sqrt{30}\sqrt{33}}\\ \cos \theta =\frac{19}{\sqrt{3\times 10}\sqrt{3\times 11}}\\ \cos \theta =\frac{19}{3\sqrt{110}}\\ \theta ={\cos }^{-1}\left(\frac{19}{3\sqrt{110}}\right)\end{array}$