Question

Find the acute angle between the two lines:
$AB$ and $CD$ passing through the points $A\equiv (3,1,-2),B\equiv (4,0,-4)$ and $C\equiv (4,-3,3)$, $D\equiv (6,-2,2)$

Answer 1

$\overline{AB}=\overline{B}-\overline{A}$

$=\hat{i}-\hat{j}-\widehat{2k}=\overrightarrow{\alpha }$

$\overline{C\Delta }=\overline{\Delta }-\overline{c}$

$=2\hat{i}+\hat{j}-\hat{k}=\overrightarrow{\beta }$

$|\overline{AB}|=\sqrt{6};|\overline{CD}|=\sqrt{6}$

$\overrightarrow{\alpha }.\overrightarrow{\beta }=|\overrightarrow{\alpha }||\overrightarrow{\beta }|\cos \theta$
$3=sqrt6.\sqrt{6}.\cos \theta$

$\frac{1}{2}=\cos \theta$

$\theta ={60}^o$