Point
Elementary
A point is an exact location, drawn as a dot. It has no width or length.
Undergraduate (affine spaces)
Consider an affine space, which has locations but no chosen origin. A point is an element of an affine space. The difference between two points is a vector (the displacement between them), and adding a vector to a point gives a point, but two points cannot be added.
Examples
Examples
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The location of Paris
Elementary
The location of Paris is a point.
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$(2, 3) \in \reals^2$
High school
$(2, 3)$ is a point of $\reals^2$, because it is a location in the plane.
Distinctions involving this term
- Vector vs. point: Points and vectors are both written as lists of coordinates in $\reals^n$, but they behave differently. A point is a location. A vector is a displacement, defined by the operations of addition and scaling. The difference of two points is a vector. Two vectors can be added, and a vector can be added to a point, but the sum of two points is not meaningful.