Question

If $A(2,1,3),B(5,3,9),C(1,-1,3)$, and $D(2,3,11)$ Find the angle between the lines $AB$ and $CD$.

Answer 1

Vector AB $A(2,1,3);B(5,3,9)$

$\overrightarrow{AB}=(5-2,3-1,9-3)$

$=3\hat{i}+2\hat{j}+6\hat{k}$

vector CD $C(1,-1,3);D(2,3,11)$

$\overrightarrow{CD}=(2-1,3-(-1),11-3)$

$=\hat{i}+4\hat{j}+8\hat{k}$

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

$=\frac{\left(3\right)\left(1\right)+\left(2\right)\left(4\right)+\left(6\right)\left(8\right)}{\sqrt{9+4+36}\sqrt{1+16+64}}=59/63$

$\theta ={\cos }^{-1}\left(\frac{59}{63}\right)$.