what is the proof for the distributive law for dot product vectors?

If vector A = (a1, a2.... , an)

then A "dot" B = a1*b1 + a2*b2 + ... + an*bn

We want to prove that A"dot" (B+C) = A"dot"B + A"dot"C

A "dot" (B+C) = a1*(b1+c1) + a2*(b2+c2) + ... + an*(bn+cn)

which we can rearrange to be a1b1 + a1c1 + a2b2 + a2c2 + etc..

Which we can further rearrange to be a1b1+a2b2 + ..+ anbn + a1c1 + a2c2 + .. + ancn = AdotB + AdotC

Dot Product Distributive

write it out, it just follows from the distributive law of ordinary numbers.