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When to use "on" and "in" prepositions in math:

  • Point on/in the plane

  • Point on/in the cartesian coordinate system

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    WRT points and planes, they are interchangeable. WRT coordinate systems, use "in" because the coordinate system does not have an identifiable surface on which a point could exist. – Victor Bazarov Oct 19 '15 at 16:25
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ON

We can describe features on surfaces or boundaries. Those surfaces can be of any dimension. (A point ON a line, a point ON a circle, a point ON "the plane," a point ON the surface of a cylinder, a point ON the surface of a klein bottle.)

INSIDE

For closed surfaces, we can refer to points that are on one side of the surface or the other (within boundaries). (A point INSIDE a circle, a point INSIDE a cylinder. A point WITHIN/INSIDE the boundaries of a region)

IN

When we refers to systems or spaces we use in. (A point IN 3-space. A point IN a coordinate system. A vector IN Hilbert Space)

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