I have two questions about the normal form of a line that I hope someone can help clarify:

1. When considering the normal form of a line, does this line need to be a line that doesn't cross through the origin? Based on this definition below, it appears that it can't cross the origin, but logically, I don't see why you would have such a restrictive definition

2. In my linear algebra textbook it says that "In R^2, only lines have a normal form, and in R^3 only planes have a normal form." I could theoretically imagine a line in R^3 that has a normal line to it, so then why does it says that in R^3 only planes can have a normal form?