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The dot product of two vectors is defined this way:

$$\begin{pmatrix} a_1\\ a_2 \\ \end{pmatrix}\cdot \begin{pmatrix} b_1\\ b_2\\ \end{pmatrix} = a_1\cdot b_1 + a_2\cdot b_2$$

I know it works exactly like it should, in the Work formula, in physics, like: $$W = F.D$$ For one-dimensional motion, only. If we need to extend this to two dimensions, we could calculate the horizontal work and the vectical work, then sum the two, like:

$$W_x = F_x\cdot Dx$$ $$W_y = F_y\cdot Dy$$ Then: $$W = W_x + W_y = F_x\cdot D_x + F_y\cdot D_y $$ If we define the dot product to work for the work formula even for a vector notation, like: $$W = F\cdot D \tag{both F and D are vectors in this case}$$ Then we must define the vector product as being the way it is.

PS:I don't explanations like 'dot product tells how much the vector is walking in certain way' because then the multiplication $a_1 \cdot b_1$ don't make sense, it could be $a_1+b_1$ and then I would still have an idea of 'how much the vector is changing'.

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If you want to calculate the angle two vectors with same origin make, then you can apply the laws of cosine on the vectors with the difference of the two vectors as the third side. When you work that out and solve for $cos (angle)$ then the numerator is exactly that dot product. The denominator happens to be always positive and so the numerator also determines whether vor not the angle is obtuse. I think that is why they call it the dot product by definition. I believe that's how I got introduced to the dot product the first time. The fact that if the dot product is zero results into orthogonal vectors is then a mere consequence.

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