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A Cartesian coordinate plane system specifies each point uniquely in a plane by a pair of numerical coordinates. These coordinates are the signed distances from the point to two fixed perpendicular directed lines often called the x and y axis. The measurements on the axes are same units of length. The use of x and y to name is axes is common, but there are many other ways to name them.
Possibly because the first time that pupils are introduced to the name and concept it is has only two dimensions. These are usually represented as horizontal (x) and vertical (y) coordinates. Actually, many children meet the concept - in its 1-dimensional form - as the number line. It is not called a Cartesian coordinate system then, and they make only simple use of it. The Cartesian coordinate system is normally extended to 3-dimensional space in high school coordinate geometry when it is obviously not called a plane, and to multi-dimensional hyperspaces in advanced mathematics or physics.
You do not have to. You could use polar coordinates, if you prefer.
Yes! By expanding the system to 3D!