Dimensions in geometry I'm having trouble understanding the concept of dimensions. Here's what wikipedia has to say:

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

Here's my problem:
Imagine a plane ABCD on a cartesian coordinate grid. Line segment CD has point E plotted on it at (1.5, 2). Given that two coordinates were required to place point E on line segment CD, shouldn't that make line segment CD a two dimensional object? Or am I looking at it the wrong way? Really this seems to be just a question of whether a line becomes a two-dimensional object if it is placed in a two-dimensional setting/space.
Here is an image to help you visualize the problem
This also has other implications elsewhere. Does a plane face on a three-dimensional hexahedron become 3D as well because 3 coordinates are needed to place a point inside the plane? And so on.
Thank-you!
 A: Notice the word 'minimum' in "the minimum number of coordinates required...". 
In the example you give you deliberately choose an inefficient way to determine the point, namely using two coordinates since the line lives in an ambient plane. But, you could do better. Choose any arbitrary point on the line, call it $p_0$. Then you can describe any other point on the line using just one single coordinate: its distance from $p_0$, measured positively in one direction, and negatively in the other direction. That shows you can determine each point on the line using just one coordinate, so certainly the dimension is $\le $ 1. Convince yourself that you can't do it with less than one coordinate, and you'll get an intuitive proof of the intuitive fact that the line has dimension $1$. 
A: You can specify a point on a line with 1 number the distance from the start of the line. The line is embedded in a 2d space so you can also specidy a point with 2 numbers. However we want the minimum number of numbers needed to specify the pointm therefore the line has dimension 1.
A: Points do not have dimension. Spaces do. So your point $E$ belongs to a plane, thus it can be represented by two coordinates. But the same point happen to belong to a line, so it could be represented by a single coordinate referenced to a basis of the one-dimensional space. And, starting from that, you have the whole topic of linear algebra in front of you...
