The dot product of two vectors returns a number that happens to be scalar. It is a representation of how two vectors are associated with each other.
Geometrically, the dot product of two vectors x and y would be as follows:
x . y= ||x|| ||y|| cosθ
θ is the angle between the vector x and y.
However, algebraically, we get the following:
Geometrically, we get the following:
θ=β-α
cosθ=cos(β-α)
cosθ = cosβ cosα + sinβ sinα
cosθ = (x1/||x||) (y1/||y||) + (x2/||x||) (y2/||y||)
||x||||y|| cosθ= x1 y1 + x2y2
x . y = x1 y1 + x2y2