# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/vector-operations/zero-vector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/vector-operations/zero-vector/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Zero Vector",
  description: "Understand zero vector properties, identity element role, and real-world applications. Learn why it's unique in vector operations and physics.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/12/2025",
  subject: "Vector and Operations",
};

## Understanding Zero Vector

A zero vector is a vector with zero length and no specific direction. Zero vector can also be defined as a vector whose initial and terminal points are at the same location. Graphically, a zero vector is represented by a single point.

Zero vector is denoted by <InlineMath math="\vec{0}" /> or 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. Zero vector is the identity element in vector addition operations. If <InlineMath math="\vec{a}" /> is any vector, then <InlineMath math="\vec{a} + \vec{0} = \vec{a}" />

2. The magnitude (length) of zero vector is zero: <InlineMath math="|\vec{0}| = 0" />

3. Zero vector cannot be normalized, as it is impossible to create a unit vector from zero vector.

4. Zero vector is unique in a vector space.

5. Zero vector is simultaneously parallel and perpendicular (orthogonal) to all other vectors.

6. The scalar product of zero vector with any real number will still result in zero vector: <InlineMath math="k \cdot \vec{0} = \vec{0}" />

## Representation of Zero Vector

In Cartesian coordinate form, zero vector can be expressed as:

- In 1-dimensional space: <InlineMath math="\vec{0} = (0)" />
- In 2-dimensional space: <InlineMath math="\vec{0} = (0, 0)" />
- In 3-dimensional space: <InlineMath math="\vec{0} = (0, 0, 0)" />

## Examples of Zero Vector in Daily Life

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

## Zero Vector in Vector Operations

### Addition with Zero Vector

If <InlineMath math="\vec{a}" /> is any vector, then:

<BlockMath math="\vec{a} + \vec{0} = \vec{a}" />

This shows that zero vector acts as the identity element in vector addition operations.

### Subtraction with Zero Vector

For any vector <InlineMath math="\vec{a}" />:

<div className="flex flex-col gap-4">
  <BlockMath math="\vec{a} - \vec{0} = \vec{a}" />
  <BlockMath math="\vec{0} - \vec{a} = -\vec{a}" />
</div>

### Scalar Multiplication with Zero Vector

For any scalar <InlineMath math="k" />:

<BlockMath math="k \cdot \vec{0} = \vec{0}" />

## Applications of Zero Vector Concept

The concept of zero vector is very important in solving various vector problems. Here are some applications:

1. In systems of linear equations, zero vector indicates a trivial solution.
2. In physics, zero vector indicates an equilibrium state, for example when the resultant force on an object is zero, the object will remain at rest or move with constant velocity (Newton's First Law).
3. In computer graphics programming, zero vector is used as a reference for coordinate transformations.