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, $$\vec a \bullet \vec b= |a||b| \cos \theta= a_1 b_1 + a_2 b_2+a_3 b_3?$$
$ a\cdot b = \|a\|\|b\|\cos\theta$ falls out from the laws of cosines.
Law of cosines says $\|(a-b)\|^2 = \|a\|^2 + \|b\|^2 - 2\|a\|\|b\|\cos\theta$
but $\|(a-b)\|^2 = (a-b)\cdot (a-b) = a\cdot a + b\cdot b - 2 (a\cdot b) = \|a\|^2 + \|b\|^2 - 2 (a\cdot b)$
Subtracting one from the other gives $0 = 2(a\cdot b)- 2\|a\|\|b\|\cos\theta$
Where it is useful? To common uses:
To show that two vectors are perpendicular.
To find the projection of one vector onto the other. That is break a vector into component vectors that are in the same direction and in an orthogonal direction.
