Three direction cosines of the line $6x - 2 = 3y + 1 = 2z - 2 $, are

$\frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}$

The equation of the line is 
$6x - 2 = 3y + 1 = 2z - 2 $

$or, \,\, 6\left(x-\frac{1}{3}\right)= 3\left(y+\frac{1}{3}\right)= 2(z-1)$

$or, \,\, \frac{x-1/3}{1}=\frac{y+1/3}{2}=\frac{z-1}{3}$

So, direction ratios of the line are proportional to 1, 2, 3.

∴ Direction cosines of the lines are

$\frac{1}{\sqrt{1+4+9}}.\frac{2}{\sqrt{1+4+9}},\frac{3}{\sqrt{1+4+9}}\,\,\, or \,\,\, \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}$