drawing using GL_LINES is old school fixed function pipeline and it's how modern graphics hardware works. If you want a single line, draw a small rectangle between V1 and V2 using geometry. The thickness is the distance between P1 and P2 / P3 and P4 of the rectangle. A line has no thickness as it's 1 dimensional.

Draw in screen space based on projected points in world space.

set gl_Color to your desired color vec and bam, line.