The vector equation of a line through point A (position vector a) parallel to vector b is r = a + lambdab, where lambda is a parameter. Every value of lambda gives a unique point on the line. The line through two points A(a) and B(b) is r = a + lambda(b - a) = (1-lambda)a + lambdab. For lambda = 0 we get A, for lambda = 1 we get B, and other values extend the line beyond AB.