# How to prove: (u+v).w=u.w + v.w ? Geometry?

How do i prove: (u+v).w=u.w + v.w ?

for all R^n. Where u,v and w are vectors R^n and the ''.'' is dot product.

Thanks very much

Update:

Also, how do i prove this one aswel?

(λu).v=λ(u.v) ?

Many thanks

Relevance

let u = (u1,...,un) v = (v1,..., vn) and w = (w1,..., wn)

then use the definitions

u+v = (u1+v1,..., un+vn)

and

λu = (λu1,..., λun)

and

u.v = u1v1+....+ unvn

I'll do the 2nd one and leave the 1st one for you

(λu).v

= (λu1,..., λun).(v1,..., vn)

= (λu1)v1+....+ (λun)vn

= λ(u1v1)+....+ λ(unvn) .....by scalar associativity

= λ(u1v1+....+ unvn) .........by scalar distributive rule

= λ(u.v)

,.,.,.

• In mathematics axioms are only assumed true for the purpose of seeing what results from that assumption. There is no assumption made based on observations of the real world. Nor is there any assumption that those axioms are themselves true in any sense. Mathematics is instead essentially those necessary truths of the form "If X then Y" where X consists of sets of axioms and Y consists of necessary conclusions based on those axioms. Historically The church fought for years against non-euclidean geometry. So much so that the introduction of non-euclidean geometry was delayed because mathematicians (namely Gauss ) feared for their saftey.