In Fisher's LDA, what is $a_i =w^T x_i$ if $w$ is an unit vector and $x_i$ is an observation?

In Fisher's LDA, what is $a_i = w^T x_i$ if $w$ is an unit vector and $x_i$ is an observation?

My notes write that:

projection lengths of data into it (unit vector $w$) as $a_i = w^T x_i$

So is $a_i$ some length or what?

Or perhaps it's a reduction of $x_i$ into lower dimension?

http://www.csd.uwo.ca/~olga/Courses/CS434a_541a/Lecture8.pdf

gives on p. 5 that $w^T x_i$ would be a scalar, which is "the distance of projection of $x_i$ from the origin.

But I have difficulties in recalling, what is this in standard vector algebra terms?