A line has a length and a height and a plane has also a depth. I don't doubt the truth but am trying to wrap my mind around this. To be more clear, if I have a line of a certain length, not parallel to either the x or y axis, a point moving in either direction will change position relative to both the x and y axis? Same with a plane and the z axis.
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$\begingroup$There are a number of variously rigorous ways to define dimension.
One thing that might influence you is the parametric form of a line:
- In two dimensions, you might write down a line as the set of all points $(x,y)$ of the form $(a_1 + b_1 t, a_2 + b_2 t)$ where $t \in \mathbb R$.
- In three dimensions, you might write down a line as the set of all points $(x,y,z)$ of the form $(a_1 + b_1 t, a_2 + b_2 t, a_3 + b_3 t)$ where $t \in \mathbb R$.
- The same kind of formula will give you a line in any number of dimensions. You can also rewrite this in vector notation as the set of points of the form $\mathbf a + t \mathbf b$ for any $t \in \mathbb R$ (where $\mathbf a, \mathbf b \in \mathbb R^n$ specify a location and direction for the line).
All of these are one-dimensional because there is a single variable $t$ that varies along the line.
On the other hand, one way to parametrize a plane (say, in three dimensions) is as the set of all points $(x,y,z)$ of the form $(a_1 + b_1 s + c_1 t, a_2 + b_2 s + c_2 t, a_3 + b_3 s + c_3 t)$ where $s,t \in \mathbb R$. Or in vector notation (and any number of dimensions): the set of points of the form $\mathbf a + s \mathbf b + t \mathbf c$ where $s,t \in \mathbb R$.
This is a two-dimensional surface because there are two variables $s$ and $t$ that vary along the plane.
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