Math Is Fun Forum

NoSash

Member

Registered: 2006-02-06

Posts: 7

Prove Associative Law using Vectors

How can I prove the associative law (multiplication) using vectors:
m(na) = (mn)a    where m and n are real numbers, and a is a vector?

There are three cases to prove:
m>0, n>0
m<0, n<0
m>0, n<0 and m<0, n>0

I have no idea where to start.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Prove Associative Law using Vectors

I'm going to change your scalars to k and l, since n is normally used to signify the length of a vector.

Since you don't know the dimension the vector is in, you have to use infinite sequence notation:

In R^n:

a = <a0, a1, .... an>

Then:

k(la) = k(<la0, la1, .... lan>) = <kla0, kla1, ... klan>

But

l(ka) = l(<ka0, ka1, ... kan>) = <kla0, kla1, ... klan>

And are thus, equal.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."