Take the given vector by itself, . Scaling the vector by a scalar will stretch the vector. Any real number can take the place of ; when we consider all possible choices, we see that traces out an infinite line through the origin and the point (2, 1, -1). (Left)
Next, treat point P(2, 1, -3) as a vector, . (Upper right) By adding to , we get a new vector that, as we change the value of , traces out the line through point P which also happens to be parallel to . (Lower right)
So the equation of this line in parametric form is
or in ijk notation,