Dot Product with Zero Vector

Notice that the length of each vector that went into the dot product of the question was greater than zero, but that the dot product was zero. Here is another example:

(0, 0, 0)^T  ·  (-2.3, 89.22, 0)^T   =   0(-2.3) + 0(89.22) + 0(0)   =   0

This is not a surprise (I hope). We saw the same thing with geometrical vectors.

0  ·  a  =  0

This looks obvious. The first 0 is the zero column matrix; the last 0 is the real number zero. Also,

0  ·  0   =   0

In each of these equations the zero column matrix means a column matrix of the same dimension as the other column matrix, with each element the real number zero.