If I understand correctly, the dot product of two vectors give you a scalar quantity. This value tells you how much, or what magnitude, of the vectors are in each others direction. I also understand that the dot product is the magnitude of the vector projection.
If this is the case, then why isn't the dot product of two identical vectors just the magnitude of that vector? Instead it is the magnitude squared. The vectors are identical and are therefore 100% in the direction of one another. Given my understanding of what a dot product is (the image above), why isn't this just the magnitude?
It might also be the mentioned that I understand the proofs for why it is the magnitude squared. However, I cannot see this visually. Thanks.
Best Answer
You have forgotten a key part of the dot product: The actual product. What you have marked on your drawing is the component of $\vec A$ along $\vec B$. That's part of the way to the dot product. The second part is that now you have to multiply the lengths together (and also consider the sign in case they are pointing in different directions).
If the length of $\vec B$ happens to be $1$, then this multiplication changes nothing, and the length you have marked on your drawing is indeed the dot product. But in general that's not the case.
The dot product is a generalisation of the regular product that you likely know from the beginning of elementary school. If you have a 1-dimensional vector space (so that your vectors are practically indistinguishable from the standard real numbers), then the dot product is just the regular product. When we generalise to higher dimensions, we want to change as little as possible, while still being able to always multiply two vectors and getting a number as a result. (There are other ways to generalise the product to higher dimensions, depending on which properties you want to keep, and which properties you want to change. The complex numbers and their multiplication is one example. Geometric algebra has its own product type. Even polynomial multiplication can be seen as such a generalisation.)