I don't quite see how an inner product(dot product), which produces a scalar from a pair of vectors can give rise to a vector which is normal to a surface.Typically only surfaces have normal vectors, lines don't. Maybe in some degenerate (can I use this word in that case?) cases, but these are special.
It generally declares the surface's direction at that point in space and is very useful in physics (involved in flux equations), math (in stereometrical geometry), mechanics (for static loads and torque calculations) and many more fields.
You will typically find it as a factor in an exterior or inner product.
We're good for the normal vectorMaybe we are a bit lost in translation and I don't vow for it this time. Let's take things from the beginning:
Is this what a normal vector is about?
http://en.wikipedia.org/wiki/Normal_vector
If the answer is yes, then I 'm in a good way.
Is this an example of an inner product?
\(\Phi _m=A \cdot B\) - Magnetic flux through a surface
(an example from this page http://en.wikipedia.org/wiki/Magnetic_flux)
If yes I 'm still on a good track.
Is this an example of an exterior product?
\(\hat{n} \times (\overline{E_2} - \overline{E_1})\) - identity for the electrical field on separating surface
If yes then I don't see any problem with my post.
If there's a no in the above, please elaborate and give me the correct translation.
The dot product of the normal vector (to a surface) with some vector field gives the flux of that vector field through the surface. Even though it is a scalar, it has a sign (+ or -). If it's positive then the flux is outward from the surface, and if negative then the flux is into the surface.We're good for the normal vector
We're good for the dot product or inner product except for the fact that it is a SCALAR quantity, not a VECTOR quantity. As a scalar quantity there is no direction and thus ther is no sense of direction -- normal parallel or otherwise.