addition (of vectors)
Given vectors a and b, let and be directed line segments that represent a and b, with the same initial point O. The sum of and is the directed line segment , where is a parallelogram, and the sum a+b is defined to be the vector c represented by . This is called the parallelogram law. Alternatively, the sum of vectors a and b can be defined by representing a by a directed line segment and b by where the final point of the first directed line segment is the initial point of the second. Then a+b is the vector represented by . This is called the triangle law. Addition of vectors has the following properties, which hold for all a, b and c:
(i) a+b=b+a, the commutative law.
(ii) a+(b+c)=(a+b)+c, the associative law.
(iii) a+0=0+a=a, where 0 is the zero vector.
(iv) a+(−a)=(−a)+a=0, where −a is the negative of a.