How does "$t$" disappear when finding the distance from a point to a line?

Question

I am trying to see why the "t" disappears when finding the distance from a point to a line in the explanation on wikipedia under the section called Vector Formulation on this page:

https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line

On that page you can see $x= a+ tn$ is the vector, and $p$ is the point. But then somehow the $t$ drops out. In particular, I do not understand this sentence on that page:

Then $(a -p )\cdot n)n\,$ is the projected length onto the line...

I do not understand how we got that expression, and the $t$ dropped out somehow.

Answer 1

1. $a-p$ is another notation for the translation vector which moves $p$ to $a$, in other words the vector $\overrightarrow{pa}$.
2. You can easily check the vector $\; a-p -\langle a-p,n\rangle n$ is orthogonal to $n$,hence its norm is the distance from point $p$ to the line.