Make 2d line from 2 points

Algebraic prove that line "l=[a b c]" which pass from 2 2d point v1=[x y] and v2=[x y] can be evaluated as:

l=[v1,1] x [v2,1], even it's obvious from fact that ([v1,1] x [v2,1], [v2,1])=0 and ([v1,1] x [v2,1], [v1,1])=0 as mixed product.