question about intuitions about basic stuff

Hi, i'm having trouble coming to an intuitive (spatial) understanding of what happens when a set of linear equations is written as a vector equation. For example, if you have three linear equations with two unknowns, this corresponds (geometrically) with three lines in the x,y plane ..right? However, viewed as a vector equation, we have two vectors in R^3 (multiplied by the respective variables) set equal to another vector in R^3. Now, instead of being in two dimensions, it seems like we've moved into three dimensions. Is this just a different geometric way of looking at the same thing?

Also, think about a system of two linear equations in three variables. We have two planes in three dimensions. However, viewed as a vector equation, we have three vectors in R^2 (Multiplied by the respective variables) set equal to another vector in R^2. How can we say these vectors are in R^2 when we're dealing with planes in three dimensions?

I'm using David Lay's linear algebra and its applications, if you have any page references.