( 4, 0, -3)^{T} **·** (0, -2, 0)^{T} = ?

( 4, 0, -3)^{T} **·** (0, -2, 0)^{T} = 4*0 + 0*(-2) + (-3)*0 = 0+0+0 = 0

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= 0·a

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, and each element is the real number zero.

More practice:

(-2, 5, -6)^{T}·( 1, 2, 3)^{T}= ?