VectorsInSpace

# 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

**opposite**of a vector*v*, is another vector,*-v*, with the same magnitude and direction but opposite sense. The coordinates are the opposite of*v*-coordinates.**To subtract**two vectors,

*u*and

*v*, we add

*u*and

*–v*.

The

**multiplication of a vector**(λЄ*v*by a scalar λ**R**), is another vector, λ*v*, 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

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