I'm learning about dot products, and I'm having a bit of trouble grasping why axbx + ayby + azbz = ab*cosΘ. I understand how it works in two dimensions, I think, but three is still fuzzy. This is what I came up with for two dimensions. The angle between the vectors is simply the difference between the angle between each vector and the x-axis. ab*cosΘ = ab*cos(Θa-Θb) = ab*(cosΘacosΘb + sinΘasinΘb) = a*cosΘab*cosΘb + a*sinΘab*sinΘb = axbx + ayby Now I understand intuitively that the dot product rule should work in three dimensions, since you could always orient the axes so that the two vectors lie on the x-y plane, in which case the above applies. But I'd like to see an analytical proof like the one above, except including azbz. I would feel a lot better about this if I could see that.