## Also Known as the Dot Product

The scalar product is one of two ways two of multiplying two vectors together. This method gives, as the name suggests, a scalar result.
A.B = |A||B|cos(ϑ)
Geometrically, this is equal to the length of A multiplied by the component of B that is parallel to A.

Clearly, from this definition, the dot products between the basis vectors in Cartesian coordinates (i,j and k) are given as:

i.i = 1, i.j = 0, i.k = 0

j.i = 0, j.j = 1, j.k = 0

k.i = 0, k.j = 0, k.k = 1

or, this can be written more concisely in coordinate notation as:
ea.eb= δab
where ea= i,j or k (for a = 1,2 or 3) and δab is the Kronecker delta (equal to one where a is equal to b zero otherwise).

This allows us to write the dot product between two vectors in terms of their Cartesian coordinates – avoiding the basically useless calculation of angles:

(Ax, Ay, Az).(Bx, By, Bz) = (Axi Ayj Azk).(Bxi Byj Bzk)

=Axi.Bxi Ayj.Byj Azk.Bzk 0

= AxBx AyBy AzBz

So, the dot product between two vectors is the sum of the products of corresponding components of the vectors.