Zero vector
Elementary
Think of vectors as arrows, which can be long or short. The zero vector is the arrow with length zero. It does not point in any direction and adding it using the tip-to-tail rule does not change the vector it is added to.
High school
The zero vector is the vector whose components are all zero.
Undergraduate
Consider a vector space $V$, with its vector addition. The zero vector of $V$ is its additive identity: the vector $\mathbf{0}$ with
\(v + \mathbf{0} = v \quad \text{for every } v \in V.\) In a space of functions, it is the function that is $0$ everywhere.
Examples
Examples
-
$(0, 0) \in \reals^2$, for vector addition
High school
$(0, 0)$ is the zero vector of $\reals^2$.
Non-examples
Things that may be mistaken for examples.
-
$0$
Elementary
The number $0$ is not a zero vector, because it is a number, not a vector.
Properties
- The zero vector is the only vector with length zero, and it has no direction.