I've looked around MSE and haven't found anything that's quite satisfied me. I understand the "hows" of the dot product but not the "whys" of the dot product. I'm not sure why the dot product would be useful. May someone please explain?
Also, if $\vec a=(a_1,a_2,a_3)$ and $\vec b=(b_1,b_2,b_3)$ then why does, $$\vec a \bullet \vec b= |a||b| \cos \theta= a_1 b_1 + a_2 b_2+a_3 b_3?$$