 Finally, by taking limits, he defined the result of multiplying a vector α by any scalar "x" as a vector β with the same direction as α if "x" is positive; the opposite direction to α if "x" is negative; and a length that is |"x"| times the length of α.The quotient of two parallel or anti-parallel vectors is therefore a scalar with absolute value equal to the ratio of the lengths of the two vectors; the scalar is positive if the vectors are parallel and negative if they are anti-parallel.A unit vector is a vector of length one.