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The Concise Oxford Dictionary of Mathematics

Christopher Clapham,

James Nicholson

addition (of vectors) 

Given vectors a and b, let addition and addition be directed line segments that represent a and b, with the same initial point O. The sum of addition and addition is the directed line segment addition, where addition is a parallelogram, and the sum a+b is defined to be the vector c represented by addition. 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 addition and b by addition 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 addition. This is called the triangle law. Addition of vectors has the following properties, which hold for all a, b and c:

  1. (i) a+b=b+a, the commutative law.

  2. (ii) a+(b+c)=(a+b)+c, the associative law.

  3. (iii) a+0=0+a=a, where 0 is the zero vector.

  4. (iv) a+(−a)=(−a)+a=0, where −a is the negative of a.