The Scalar Product

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:

e_a.e_b= δ_ab

where e_a= i,j or k (for a = 1,2 or 3) and δ_ab 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:

(A_x, A_y, A_z).(B_x, B_y, B_z) = (A_xi A_yj A_zk).(B_xi B_yj B_zk)

=A_xi.B_xi A_yj.B_yj A_zk.B_zk 0

= A_xB_x A_yB_y A_zB_z

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