How do I prove sa+ta=(s+t)a and s(ta)=(st)a for vectors?

[grimster]

a is a vector and s and t are two integers. I'm supposed to show that:

sa+ta=(s+t)a

and

s*(ta)=(s*t)a

the two are so obvious I'm not sure how i prove them.

 

[arildno]

Make a careful check of the axioms given for your vector space, and see what needs to be proven.

 

[grimster]

thing is, i think I'm supposed to show it geometrically by drawing it. how do i do that?

 

[*best&sweetest*]

let a = (x,y) in the component form
then
sa+ta =
s(x,y) + t(x,y) =
(sx,sy) + (tx,ty)=
(sx+tx,sy+ty)=

(x(s+t),y(s+t))=

(s+t)(x,y)=
(s+t)a and do the similar for b)

 

[arildno]

Well, sa is parallell to a, isn't it?
And so is ta..
So, how would you geometrically add these vectors, and what resultant vector does this equal?

 

[grimster]

Well, sa is parallell to a, isn't it?
And so is ta..
So, how would you geometrically add these vectors, and what resultant vector does this equal?

(s+t)*a

but that is what I'm supposed to show. so is it enough to just draw sa and then ta from where sa ends? add them together so to speak?

 

[arildno]

I guess so.

 

[Office_Shredder]

That's about as geometrically proven as it gets...