Zwiebach scalar product notation I am currently working through Zwiebach's a First Course in String Theory.
He seems to use dot-product notation interchangeably with the "down-up" notation.
For example, on pg 176/section 9.1, he talks about the "gauge condiction $n\cdot X = \lambda \tau$ fixing the string...", "points $X^\mu$ that satisfy $n_\mu X^\mu=\lambda \tau$ are points that lie both on the world-sheet and on the hyperplane...", "the string with world-sheet time tau is the intersection of the world-sheet with the hyperplane $n \cdot x = \lambda \tau$ as illustrated...", and the illustration has an arrow pointing to said hyperplane with the $n_\mu x^\mu = \lambda \tau$ label.
Does anybody know why he does this?
 A: 
Disclaimer: I've never read this particular book, but I'm reasonably certain index notation is invariant across most of physics.

The mathematical notion of interest is of course the inner product in some, well, inner product space. The definition of an inner product is given by a bilinear form - the metric. Let's call it $g$. Then
$$ (a, b) \equiv g(a, b) \equiv g_{\mu\nu} a^\mu b^\nu \equiv a_\mu b^\mu. $$
In pure math, as often as not, writing $a \cdot b$ implies the metric on hand is Euclidean, and all we are doing summing is over elementwise products of the numbers. In physics, on the other hand, it is quite common for $a \cdot b$ to mean the inner product with respect to whatever metric we have on hand.
As for why both notations are in use simultaneously: I'd speculate that dot notation is meant to evoke a geometric intuition, while index notation is is easier to actually do computations with. (Though I'd argue that one should be familiar enough with the latter to instantly recognize geometric notions encoded in it.)
