Creating a function that performs a "`DotExpand`"

Question

When you make a Dot[a,(a+b+c)] the output will be:

a.(a + b + c)

I want to make a function that expands the dot product, for the previous example would be:

a^2 + a.b + a.c

I don't have much knowledge about pure function. I start to create the function as following:

DotExpand[Dot[l_, r_]] := Map[Times[#, #] &, {l, r}]

In[36]:= DotExpand[(a + b + c).a]

Out[36]= {(a + b + c)^2, a^2}

As you see, is not working. What I have do modify to works?

Answer 1

Not sure, are you looking for something using Distribute?

DotExpand[u_,v_]:=Distribute[Dot[u,v]]

For example,

DotExpand[a,a+b+c]

a.a + a.b + a.c

DotExpand[a+b,b+c]

a.b + a.c + b.b + b.c