I'm reading a differential geometry book named "Elementary geometry of differentiable curves: An undergraduate introduction" written by Gibson.
I'm on page 2. On that page, 4 properties of scalar product(dot product) of vectors are given.
One of them is:
I know the proof of this and the proof shows that the statement is true for
I accept this.
But if isn't it definite that has to be equal to and not greater than ?
Can anyone find a such that when then is greater than ?
Why did the author said that ? Why did he mention greater and equal, and not just equal to ?
it's just a concise of saying:
if z ≠ 0, z.z > 0
if z = 0, z.z = 0.
from these 2 statements, we can conclude that if z.z = 0 (that is, z.z is not greater than 0), then z must itself be 0 (for if it were non-zero, then z.z > 0).
in general, all we can say is that z.z ≥ 0, for an arbitrary vector z, because of the fact that z might be 0.
it's like saying: "a square (of a real number) is always positive"....except for 0. Zero-objects are funny like that, they often introduce "exceptions" which make simple rules more complicated to say.