operations with vectors

To add two vectors, u and v, we join the extreme of u with the origin of v and then, u + v has the origin of u as its origin and the extreme of v as its extreme.

The multiplication of a vector v by a scalar λ (λЄR), is another vector, λv, with:

 

–magnitude:|λ|·|v|

 

–the same direction with the same sense if λ > 0, and opposite sense if λ < 0.

 

 

PROPERTIES: let u, v, w free vectors and λ, µ real numbers

 

(i) Commutative property: u + v = v + u

 

(ii) Associative property: u + (v + w) = (u + v) + w

 

(iii) Additive identity:

 

(iv) Additive inverse:

 

(v) Distributive properties:

     (λ +µ)·u=  λ·u + µ·u              λ(u+v) = λ·u + λ·v 

(vi)

 

(vii) Triangle inequality:  |u + v| ≤ |u| + |v|

 

With all these properties , 

  has a vector space structure