In a video lecture (MIT 6.034 Artificial Intelligence, Fall 2010 lecture 16)
It was mentioned that the dot product here is taking the projection of $\vec u$ onto $\vec w$ and it should be greater than some constant. $\vec w$ is the normal to the decision boundary
$\vec w.\vec u>c$
and to generalize it if $\vec w.\vec u+b>0$ positive sample negative otherwise
but as far as I remember length of the projection is
$\frac{|\vec w.\vec u|}{|\vec w|}$
but here a different equation is used
It is correct that $|\vec w \cdot \vec u|/|\vec w|$ is the length of the projection of $\vec u$ onto $\vec w$. If $\vec{w}$ is a unit vector, i.e. $|\vec{w}| = 1$, then this becomes simply $|\vec w \cdot \vec u|$. In general, if $\vec{w}$ is not a unit vector, then you are right to point out that $|\vec w \cdot \vec u|$ is not exactly the length of the projection of $\vec u$ onto $\vec w$, since it differs by a factor of $|\vec w|$. However, you could think of this factor as being absorbed into $c$ on the right side of the inequality $\vec w \cdot \vec u > c$, so in the end the discrepancy does not matter.
