Vectors question.

[GiantEvil]

When adding 2 vectors by the parallelogram method, it seem's obvious that the operation is always commutative, although I'm not sure how to go about constructing a proof of this.

When adding >2 vectors is the operation always distributive?

[timo]

Distribution is not a property of addition but a property of a combined addition and multiplication. You'd have to say what your multiplication is.

 

In mathematics one defines a vector as an element of a vector space, which is a set of elements (that do not need to the the tuples (x,y,z) that you are familiar with but can also be more abstract) that adhere to certain calculation rules. Among these rules is that v1 + v2 = v2 + v1 (commutativity) and x*(v1 + v2) = x*v1 + x*v2 where x is a "scalar" (think of it as a real number). This latter rule is of course a form of the distributivity. So in some sense all vectors adhere to some distributivity by definition. But you possibly need to be a bit careful what you mean by that.

 

In case you meant "associative" instead of distributive: yes, vector addition is associative.

[DrRocket]

When adding 2 vectors by the parallelogram method, it seem's obvious that the operation is always commutative, although I'm not sure how to go about constructing a proof of this.

When adding >2 vectors is the operation always distributive?

 

Vector addition is always commutative. Scalar multiplication distributes over vector addition.

 

All of this comes from the definition of a vector space, and is more general than the "parallel law". The parallel law, and other geometric aspects of vector addition come from additional structure that can be imposed, such as a norm or an inner product.

 

You can show commutativity of "parallelogram addition", by drawing pictures, which is basically how the operation is described.

 

The definition of a vector space starts with an abelian group of vectors, hence commutativity of vector addition, coupled with an operation of scalar multiplication that is required to have the usual properties of associativity and distributivity. See any text on abstract algebra or linear algebra. Volume 3 of Nate Jacobson's three volume set Lectures in Abstract Algebra is one good source.

[mathematic]

When adding 2 vectors by the parallelogram method, it seem's obvious that the operation is always commutative, although I'm not sure how to go about constructing a proof of this.

When adding >2 vectors is the operation always distributive?

 

I think you mean "associative" not "distributive". The answer is yes. As for proofs, note that the addition of vectors can be viewed as separate additions for each component, where the commutative and associative laws hold (ordinary arithmetic).

[GiantEvil]

DOH! Yeah I meant "associative", but hey, "distributive" was an interesting question as well.

Just out of curiosity, what is an applied example of multiplying scalars? That is, in modeling what situation, would a person use that?

[DrRocket]

DOH! Yeah I meant "associative", but hey, "distributive" was an interesting question as well.

Just out of curiosity, what is an applied example of multiplying scalars? That is, in modeling what situation, would a person use that?

 

 

Channge of scale.

[GiantEvil]

Channge of scale.

DOH! Well, thanks for the answer.