TITLE: How exactly does the sign of the dot product determine the angle between two vectors?
QUESTION [12 upvotes]: 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.

REPLY [7 votes]: 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.