Vector
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
Examples
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$(2, 3) \in \reals^2$
High school
$(2, 3)$ is a vector, because $\reals^2$ has a distinguished origin, $(0, 0)$, so each point can be identified with the vector from the origin.
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$(3, 2)$ is the coordinate vector of $(3, 2) \in \reals^2$, relative to the standard basis
High school
$(3, 2)$ is a vector in $\reals^2$, because it is an element of the vector space $\reals^2$.
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A velocity, in meters per second
High school
A velocity is a vector because it is an arrow in space, with units of length per time.
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The polynomial $p(x) = 3 + 2x$
High school
$p(x) = 3 + 2x$ is a vector in the space of polynomials of degree 1 or less.
Non-examples
Things that may be mistaken for examples.
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The location of Paris
Elementary
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.
Distinctions involving this term
- Vector vs. covector: Vectors and covectors are both described by lists of numbers in coordinates, so they are easy to confuse. A covector is a linear function that takes a vector and returns a number. In coordinates, both look like lists of numbers, but when the basis changes, their coordinates change in opposite ways.
- 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.
- Vector vs. coordinate vector: Vectors in a vector space are often described by lists of numbers, which depend on a choice of basis. A vector is an element of a vector space. Its coordinate vector is a list of numbers (coordinates) that describes it relative to a chosen basis. Changing the basis changes the coordinates, not the vector.