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Basically i am trying to understand vector multiplication, i dont know if it is the cross product or the dot product i am thinking of

Okay so here is the question and what is confusing me in the answer

So if we have two vectors and we multiply them

a.b this in my mind as i understand it means this(in 2d):

(a_x + a_y ) * (b_x +b_y) = a_x * b_x + a_x * b_y + a_y * b_x + a_y * b_y

now i dont understand why the dot product misses these in the centre? and goes straight to only a_x * b_x + a_y * b_y (or the other way a.b.cos(theta))

what is it exactly that is being multiplied here? and furthermore what exactly is being found here?

if i wanted to move a vector from position a -> position b , by using the dot product am i finding that space in between? (i.e the vector which is required to be added to a to transform vector a to b?)

its really confusing me all of this?