# addition

# 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**.