If you draw a straight line.. it is 1 dimension says the math community. But if so... Which plane is that 1 line drawn? The xy plane?, The xz plane? Or the yz plane?

And if one of those three 2D planes is our only option what stops a 2D object being hiding behind that 1D line?
>But if so... Which plane is that 1 line drawn? The xy plane?, The xz plane? Or the yz plane?

Yes, those are some of the infinitely many planes in which you could draw a line.

>And if one of those three 2D planes is our only option what stops a 2D object being hiding behind that 1D line?

I'm not sure I understand what you are trying to ask here.

Just like a 3D object can hide behind a 2D one, a 2D object can hide behind a 1D one when oriented correctly. Nothing stops it; it can happen.

And again, those three planes are not the only options. There are infinitely many different planes in R^(3), and infinitely many lines in each of those planes.
A line is a 1 dimensional object because it can be formed from scalar multiples of a single point (sometimes called a vector) in 2-space, 3-space and so on.

Imagine what would happen when you take the coordinates (ta,tb,tc) in 3-space and let t vary through the real number line while a,b,c are constant real numbers. It’s not hard to imagine that if we were to plot this set of points for every t in the real numbers we’d get a straight line through the origin. Yet, we need to vary only a single value. In this sense the number of values we must vary, also called parameters, to write our object in coordinate form like this gives you the dimension.
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It depends on the line.  It can be the intersection of two of those planes, or of others of the infinite planes that exist.

Edit: For some reason, people are downvoting the idea that a line is the intersection of two planes in three-space.  Splendid.