Vector vs. point
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.
Definitions
Level:
Elementary
A vector is an arrow with a length and a direction.
High school
A vector is a quantity with a magnitude and a direction. Graphically, a vector is drawn as an arrow. In coordinates, it is written as a list of numbers.
Caveat. In general vector spaces, such as spaces of functions or polynomials, there may be no meaningful notion of “magnitude” or “direction” until one is chosen.
Undergraduate
A vector is an element of a vector space: a set with vector addition and scalar multiplication satisfying the vector space axioms. Vectors are defined by what you can do with them, not by what they look like.
Examples
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$(2, 3) \in \reals^2$
$(2, 3)$ is a point of $\reals^2$, and it is a vector, because $\reals^2$ has a distinguished origin, $(0, 0)$, so each point can be identified with the vector from the origin.
-
A velocity, in meters per second
A velocity is a vector because it is an arrow in space, with units of length per time. It is not a covector, because it does not take a vector and return a number.
Non-examples
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The location of Paris
The location of Paris is a point. It is not a vector, because adding the locations of Paris and the location of another city, such as Rome, has no meaning.
Missing definition: point
Examples
| Example | vector | point | Why |
|---|---|---|---|
| The location of Paris | ✗ | ✓ | The location of Paris is a point. It is not a vector, because adding the locations of Paris and the location of another city, such as Rome, has no meaning. |
Missing example: the_displacement_from_paris_to_rome | |||
| $(2, 3) \in \reals^2$ | ✓ | ✓ | $(2, 3)$ is a point of $\reals^2$, and it is a vector, because $\reals^2$ has a distinguished origin, $(0, 0)$, so each point can be identified with the vector from the origin. |
Notes
- An affine combination of points, $\lambda p + (1 - \lambda) q$, is meaningful because it equals $q + \lambda (p - q)$. For example, the midpoint of $p$ and $q$ is $\frac{1}{2} p + \frac{1}{2} q$.
- In physics, positions are points and velocities are vectors. Changing the origin changes the coordinates of positions but not of velocities.