Null Vector

TL;DR

• A vector whose components are all zero and therefore has no direction.
• Adding or subtracting it leaves any other vector unchanged.
• Commonly used as the origin or reference point in coordinate systems and appears as the trivial solution in linear equations.

Definition

A null vector, also known as a zero vector, is a vector with a magnitude of zero. It is typically represented by a vector whose components are all zero.

Explanation

A null vector has no direction or magnitude because every component equals zero. As a result, when it is added to or subtracted from another vector, the other vector remains unchanged. Null vectors serve as standard reference points in coordinate systems (the origin) and appear in linear algebra as the trivial solution to linear equations.

Examples

Two-dimensional null vector

The vector (0,0) has no magnitude or direction and can represent the origin in a two-dimensional coordinate system.

Three-dimensional null vector

The vector (0,0,0) is used similarly in three-dimensional space to represent the origin.

Vector addition and subtraction

Adding (2,4) and (0,0) yields (2,4), because the null vector (0,0) has no effect on the other vector. Subtracting (0,0) from (2,4) likewise returns (2,4).

Linear equation example

The linear equation

$$2x + 3y = 0$$

has the null vector (0,0) as a solution because it satisfies the equation.

Use cases

• Reference point or origin in coordinate systems.
• Identity element in vector addition (leaves other vectors unchanged when added or subtracted).
• Trivial solution in linear algebra problems and systems of equations.

Notes or pitfalls

• A null vector cannot represent a nonzero physical quantity, since it has no magnitude or direction, but it is useful as a coordinate origin or reference.

Related terms

• Zero vector
• Origin
• Coordinate system
• Vector operations
• Linear algebra
• Magnitude
• Direction