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.
Definitions
A vector is an arrow with a length and a direction.
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.
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
-
$(2, 3) \in \reals^2$
$(2, 3)$ is a vector, because $\reals^2$ has a distinguished origin, $(0, 0)$, so each point can be identified with the vector from the origin.
-
$(3, 2)$ is the coordinate vector of $(3, 2) \in \reals^2$, relative to the standard basis
$(3, 2)$ is a vector in $\reals^2$, because it is an element of the vector space $\reals^2$.
-
A velocity, in meters per second
A velocity is a vector because it is an arrow in space, with units of length per time.
-
The polynomial $p(x) = 3 + 2x$
$p(x) = 3 + 2x$ is a vector in the space of polynomials of degree 1 or less.
Non-examples
-
The location of Paris
The location of Paris is not a vector, because adding the locations of Paris and the location of another city, such as Rome, has no meaning.
A point is an exact location, drawn as a dot. It has no width or length.
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
-
The location of Paris
The location of Paris is a point.
-
$(2, 3) \in \reals^2$
$(2, 3)$ is a point of $\reals^2$, because it is a location in the plane.
Examples
| Example | vector | point | Why |
|---|---|---|---|
| The location of Paris | ✗ | ✓ | |
Missing example: the_displacement_from_paris_to_rome | |||
| $(2, 3) \in \reals^2$ | ✓ | ✓ | |
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.