# Problem: Is the line through (4,6,3) and (2,0, -1) parallel to the line through (6,22,12) and (1,7,2)?

###### FREE Expert Solution

The equation of a line with the direction vector d = (l,m,n) passing through a point (x1,y1,z1) is:

$\overline{)\frac{\mathbf{x}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}}{\mathbf{L}}{\mathbf{=}}\frac{\mathbf{y}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}}{\mathbf{m}}{\mathbf{=}}\frac{\mathbf{z}\mathbf{-}{\mathbf{z}}_{\mathbf{1}}}{\mathbf{n}}}$

For the first line:

$\frac{\mathbf{x}\mathbf{-}\mathbf{4}}{\mathbf{4}\mathbf{-}\mathbf{2}}\mathbf{=}\frac{\mathbf{y}\mathbf{-}\mathbf{6}}{\mathbf{6}\mathbf{-}\mathbf{0}}\mathbf{=}\frac{\mathbf{z}\mathbf{-}\mathbf{3}}{\mathbf{3}\mathbf{-}\mathbf{\left(}\mathbf{-}\mathbf{1}\mathbf{\right)}}\phantom{\rule{0ex}{0ex}}\frac{\mathbf{x}\mathbf{-}\mathbf{4}}{\mathbf{2}}\mathbf{=}\frac{\mathbf{y}\mathbf{-}\mathbf{6}}{\mathbf{6}}\mathbf{=}\frac{\mathbf{z}\mathbf{-}\mathbf{3}}{\mathbf{4}}\phantom{\rule{0ex}{0ex}}\frac{\mathbf{x}\mathbf{-}\mathbf{4}}{\mathbf{1}}\mathbf{=}\frac{\mathbf{y}\mathbf{-}\mathbf{6}}{\mathbf{3}}\mathbf{=}\frac{\mathbf{z}\mathbf{-}\mathbf{3}}{\mathbf{2}}$ ###### Problem Details

Is the line through (4,6,3) and (2,0, -1) parallel to the line through (6,22,12) and (1,7,2)?