The Algebraic Intricacies of Dot Products

Warm up '

A first warm-up question involves the algebra of dot products.
The dot product of the vector 2v with the vector 2w, when compared with y times w, is
equal to..?
There are two choices: one is to multiply V 2 times v dot w, or is it 4 times V dot w?
(2v->)*(2w->)=2(v->*w->) or 4(v->*w->)
The question remains. Which one is true?
The dot product of 2v with 2w is 2w1 comma 2w2 times w2.

What happens if we multiply it
out?
The product of these things is 4v1w1 and the product of these things is 4v2w2, and that
means the following.
(2v->)*(2w->)=(2v_1,2v_2)*(2w_1,2_w2)=4v_1,w_1+4v_2,w_2=4(v->*w->)
Here we see that generally, it is possible to guess what will happen.
If I take the dot product of lambda times v and w, then that is equal to the dot product of v
with w by the same reasoning.

So I already have lambda v1 and lambda v2 here. Here, I have just w1 and w2. Then we
multiply all of these out, and you get lambda times the original dot product.
(λv->)*w->=λ(v->*w->)