Tangent plane passes through origin This is from a section in my course book on elementary differential geometry:

Since the tangent plane $T_p S$ of a surface $S$ at a point $p \in S$ passes through the origin of $\mathbb{R}^3$, it is completely determined by giving a unit vector perpendicular to it...

There are plenty of surfaces with points whose tangent plane doesn't pass through the origin, so why does it say so here?
 A: I THINK:
The author identified the tangent space $T_p\mathbb{R}^3$ and $\mathbb{R}^3$ itself. In this case $T_pS$ is a plane passing through the origin i.e. a subspace of co-dimension one and hence the normal and $p$ determine the plane.
A: Look at the definition of "tangent plane of a surface $S$ at a point $p \in S$" in your textbook. As mentioned in the comments, there's a pretty good chance that it's defined to be a vector space satisfying certain conditions. If so, then it certainly contains the origin (since all vector spaces contain the origin).
Intuitively, you would expect the "tangent plane at $p$" to be the plane passing through $p$ that's tangent to the surface at $p$, in which case, as you say, it would not necessarily pass through the origin. Apparently, the guy who wrote your book didn't think it was important for the definition to conform with intuitive ideas.
The basic problem, I think, is that vector spaces are not a very good way to model planes, since this only allows for planes passing through the origin. To model arbitrary planes, you need to introduce notions of "point" and "affine space", which your author didn't want to bother with, I guess.
A: Planes in space not through the origin are vector spaces. However, not with respect to the standard vector space structure of $\mathbb{R}^n$. To give a plane containing base point $p$ a vector space structure one may simply identify $p+v$ in the plane with $(p,v)$ hence addition and scalar multiplication are given by:
$$ (p,v)+_p (p,w)  = (p,v+w) \qquad \& \qquad c \cdot_p (p,v) = (p,cv) $$
here I have denoted vector addition at $p$ by $+_p$ and scalar multiplication by $\cdot_p$ just to be annoyingly explicit about the fact these are nonstandard addition and scalar multiplication. This formality I give here is nothing more than the standard tip-to-tail vector addition we teach in elementary courses paired with our insistence that you can "move" the vectors around. When we make such statements it implicits the fact that we are not really working with vectors based at the origin. Instead the tail of the vector may be placed any old place in space. For this reason, it is important to distinguish language such as the vector points "to the plane" verses the vector lies "on the plane" or "in the plane". Much as we might like to avoid it, this nonstandard addition is peppered throughout the introductory and popular uses of vectors. That's my take. Perhaps the author had in mind this structure.
A: Probably, your book defines $T_p S$ as the set of the tangents of all the curves $\alpha:I \longrightarrow S$, which is a 2-dimensional vector space (i.e. a plane that passes through the origin).
