- Prove that for all vectors in we have commutativity of vector addition,
associativity of vector addition,
associativity of scalar multiplication,
and distributivity,
- Illustrate the geometric meaning of the distributive laws
- Proof. Let , , and be vectors in and let be scalars. Then by commutativity of addition in we have
By associativity of addition in we have
For associativity of scalar multiplication we have
For the first distributivity law we have
For the second distributivity law we have
- The distributive law means the vector is obtained by adding the arrow to the end of the arrow .
The distributive law means that the vector is the vertex of the parallelogram formed by and .