A plane is 2-dimensional, and has infinite extent in any direction.
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The plane that geometry refers to is not an airplane. A plane is a two-dimensional group of points that goes on infinitely in all directions; made up of infinite line. For example, a rectangle is a plane.
fg
A line.
A straight line
either the x plane or the y plane by themselves is one dimensional. Also any intersection of the x and y planes as described as sets or by an equation are one dimensional. In order to become three dimensional an intersecting plane, such a "z" plane would have to be added.