Zero Vector

A Vector Whose Endpoints Coincide

The zero vector is a vector with magnitude zero and no defined direction. Its initial and terminal points coincide, so its arrow collapses to a single point in a geometric drawing.

The zero vector is denoted by $\vec{0}$, the number $0$ with an arrow above it.

Properties of Zero Vector

Zero vector has several special properties that distinguish it from other vectors:

1. The zero vector is the identity element for vector addition. If $\vec{a}$ is any vector, then $\vec{a} + \vec{0} = \vec{a}$

2. Its magnitude is zero: $|\vec{0}| = 0$

3. It cannot be normalized because dividing by its zero magnitude is undefined.

4. Every vector space has exactly one additive identity, so its zero vector is unique.

5. For every vector $\vec{v}$, $\vec{0} \cdot \vec{v} = 0$, so the zero vector satisfies the algebraic dot-product condition for orthogonality. Because it has no direction, however, it should not be described as geometrically parallel to a direction.

6. Multiplying it by any real scalar still gives the zero vector: $k \cdot \vec{0} = \vec{0}$

Representation of Zero Vector

Every Cartesian component of the zero vector equals zero, so its length is zero and it has no direction. The list below shows the coordinates in one, two, and three dimensions.

• In $1$-dimensional space: $\vec{0} = (0)$
• In $2$-dimensional space: $\vec{0} = (0, 0)$
• In $3$-dimensional space: $\vec{0} = (0, 0, 0)$

Returning to the Starting Point

One example of a zero vector in daily life is the displacement of someone who returns to their starting position. For instance:

If Andi walks $100 \text{ m}$ east and then $100 \text{ m}$ west, his total displacement is zero. Although Andi has traveled a total distance of $200 \text{ m}$, his displacement (the difference between final and initial positions) is zero.

Zero Vector in Vector Operations

Vector arithmetic needs one element that leaves every vector unchanged, and the zero vector fills that role. Adding it changes nothing, and subtracting it only flips the sign.

The zero vector behaves like the number zero in vector arithmetic.

Addition with Zero Vector

If $\vec{a}$ is any vector, then:

The zero vector is therefore the identity element for vector addition.

Subtraction with Zero Vector

For any vector $\vec{a}$:

Scalar Multiplication with Zero Vector

For any scalar $k$:

Applications of Zero Vector Concept

The zero vector appears in the following settings:

1. In a homogeneous system of linear equations, the zero vector is always the trivial solution.
2. In physics, a zero resultant force on an object with nonzero mass means zero acceleration. In an inertial frame, the object therefore remains at rest or continues with constant velocity, as described by Newton's first law.
3. In computer graphics, the zero vector represents no displacement and therefore leaves a position unchanged when used as a translation vector.