3.7.5—Whether two lines are parallel

Two vector lines are parallel when their direction vectors are scalar multiples. They are identical only if they also share a point; otherwise they are distinct parallel lines.

Compare direction components first, then test a point from one line in the other. In 3D, skew lines are neither parallel nor intersecting.

r=(1,0)+λ(2,3) and r=(4,5)+μ(−4,−6) are parallel because directions are proportional; checking the point decides whether they coincide.

Equal gradients in a 2D graph are not enough to prove coincident lines; intercepts matter.