The "norm" of a vector, || v ||, has been defined:

    (1) as the vector's length
    
    (2)  as the square root of the sums of the squares of the vector's elements, that is:

             

We have also seen two formulas for the dot product:

    (1) u . v = || u || || v || cos θ
    
    (2) u . v = + + ... +
    
In both cases, I believe that formulas (1) are the fundamental ones, and that formulas (2) are simply how the relationships of the vectors' coordinate elements work out when the vectors are expressed in the standard basis (or, perhaps, in any orthogonal or orthonormal basis.)

The "coordinate based" formulas for the norm and dot products (and perhaps other similar formulas we have been using) and intended for use with coordinate vectors expressed in the Standard Basis, and not any arbitrary basis.

Created by Mathematica  (August 6, 2007)