Can you explain this simple vector algebra?

Question

This is a part of a simple derivation in my textbook, but I'm really not seeing where it came from.

$\frac{\vec{E}\cdot d\vec{s}}{|d\vec{s}|} = E \cdot \vec{n}$ where n is the normal vector to $d\vec{s}$. However, shouldn't n be the unit vector rather than the normal vector?

Answer 1

Maybe you are (reasonably) confusing the notion of normal vector with the notion of a normalized vector (dividing by its norm to make it have norm equal to one)?

Answer 2

Yes you are correct. $\frac{\vec{x}}{|\vec{x}|}$ is the unit vector in the direction $\vec{x}$, not normal to $\vec{x}$.