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**x3bnm** 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:

$\displaystyle \mathbf z \cdot \mathbf z \geq 0 \text{ with equality if and only if } \mathbf z = \mathbf 0$

I know the proof of this and the proof shows that the statement is true for $\displaystyle \geq$

I accept this.

But if $\displaystyle \mathbf z = \mathbf 0$ isn't it definite that $\displaystyle \mathbf z \cdot \mathbf z$ has to be equal to $\displaystyle 0$ and not greater than $\displaystyle 0$?

Can anyone find a $\displaystyle \mathbf z$ such that when $\displaystyle \mathbf z = \mathbf 0$ then $\displaystyle \mathbf z \cdot \mathbf z$ is greater than $\displaystyle 0$?

Why did the author said that $\displaystyle \mathbf z \cdot \mathbf z \geq 0$? Why did he mention greater and equal, and not just equal to $\displaystyle 0$?