Basically whenever I imagine a surface, by normal at a point, I mean a straight line perpendicular to the surface at that point which has an infinite length as straight lines do have. But how does this straight line have a magnitude when I observe this from a vectorial viewpoint? Then I consider the normal to be a vector and it has a magnitude. But co-ordinate geometry is analogous to vectorial geometry. That is , I can solve co-ordinate geometry problems by vectors and vice-versa. Still this thing confuses me.
Can anyone offer some help?
The straight line orthogonal to a surface at a point $P$ is ,in fact, a line so it has infinite length.
But the equation of a straight line is represented, usually, as $\vec x= \vec v t +\vec v_0$, here $\vec v$ is a vector that gives the direction of the line, and this vector has a magnitude, but this magnitude has nothing to do with a ''magnitude of the line'', an expression that has no meaning.
Note that this vector is also oriented, but the straight line is the the same for opposite oriented vectors (really for all parallel vectors), with a positive orientation that is the same of the vector.
In many application it's important to define an orientation of the normal and also to have an orienting vector that has unit magnitude, so we define a normal vector to the surface, i.e. a unitary vector that orients the orthogonal line and also fix an orientations as positive.
This is useful especially in physical application as in the definition of the flux of vector field through a surface.