In Apostol's Calculus Volume II, he introduces ten axioms that define a linear space. Thereafter, an example (example 4, page 5) is given of a linear space. This fourth example I do not understand. It states:
EXAMPLE 4. Let V be the set of all vectors in Vn, orthogonal to a given nonzero vector N. If n = 2, this linear space is a line through 0 with N as a normal vector. If n=3, it is a plane through 0 with N as normal vector.
The reason for my difficulty in understanding this example is due to the fact that I cannot visualize it. I understand (correct me if I'm wrong), that when saying n=2, we are saying that there are two dimensions and when n=3, there are three. This is why there is line in the first case, and a plane in the second. Now, more specifically my questions are:
-Is there a physical (or 'everyday' normal) example or analogy of this case above? -why is the line passing only through zero? why not anywhere else?
I am just beginning a long journey through analytic math with books like Apostol's. Any tips for such an endeavor would be wonderful as well.