Thursday, February 09, 2006

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:
    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.

No comments: