How exactly does the sign of the dot product determine the angle between two vectors?

I am told that $v\cdot w=0$ means that the angle between the vectors $v$ and $w$ is $90$ degrees. Then I am told that the sign of $v\cdot w$ (when it isn't equal to zero) determines whether the angle between vectors $v$ and $w$ is above or below $90$ degrees; The angle is above $90$ degrees when $v\cdot w<0$ and below $90$ degrees when $v\cdot w>0$

I have a picture from the book Introduction to Linear Algebra by Gilbert Strang, but it's quite confusing for me. I don't understand how the above information is reflected in this diagram. I would like an explanation of this for me as it will help me answer a question from the section's accompanying problem set.

Solution 1:

The dot product between two vectors $\vec v$ and $\vec w$ is given by: $$ \vec v \cdot \vec w = |\vec v||\vec w| \cos \theta $$ where $\theta$ is the angle ($0\le\theta\le \pi$) between the two vectors, so it is positive if $\cos \theta >0 \iff 0\le \theta < \pi/2$ and it is negative if $\cos \theta <0 \iff \pi/2 < \theta \le \pi$  

So, if $r$ is a straight line orthogonal to $\vec v$ and passing through its origin that divide the plane in two semiplanes, than the dot product is positive if $\vec w$ is in the same semiplane as $\vec v$ and is negative if $\vec w$ is in the other semiplane. And this is what is illustrated by the figure.

Solution 2:

You have the following characterization:

$$v\cdot w=\Vert u\Vert\Vert w \Vert\cos (\theta)$$

Where $\theta$ is the angle between $u$ and $w$, with $0\leq \theta \leq 180°$

From this, you can see for example that this angle lies between $0$ and $90°$ if and only if $\cos(\theta)>0$ if and only if $u\cdot w>0$ (since the norms $\Vert u\Vert,\Vert w \Vert$ are greater than zero)