Dot products or "scaler products" are the first significant manipulation we learn to use for vectors after the famous "head to tail" rule. Unlike the "head to tail" rule of vector addition and subtraction, dot products appear initially to be somewhat obscure, however, they have both a signficant geometrical and physical meaning.
Geometrically, the definition of a dot product allows the angle between to two vectors to determined quite rapidly using:
a.b = ab cos x = xaxb + yayb + zazb
a = xai + yaj + zak and b = xbi + ybj + zbk
cos x = (a.b)/(ab)
where the bold italic symbols refer to vectors, x is the angle between the two vectors and non-bold symbols refer to the magnitudes of the vector. This is a very straight forward calculation for two vectors that are defined and is much simpler than the comparable cartesian algebraic approach.
The procedure also has a physical meaning, for example, the work done (W) by a Force (F) displacing an object along a vector (r) can be calculated using:
W = F.r
In this case the dot product has a precise a physical meaning .... sounds like a good idea, simple and useful !!