The question is:
This following answer is quite vauge and I do not think I can develop it further alone, but it may it be of some help, should I post it as an answer?
Where $A$ and $B$ are vectors, and $x$ is $A ∠ B$.
$A · B = |A| |B| cos(x)$
This uses trigonometry, however the following definition achieves the same result without using trigonometry: