# Nakafa Framework: LLM

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

Output docs content for large language models.

---

import { VectorChart } from "@repo/design-system/components/contents/vector-chart";

export const metadata = {
  title: "Vector Types",
  description: "Discover zero, negative, equivalent vectors and their properties. Learn magnitude, direction concepts with interactive visualizations and exercises.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/08/2025",
  subject: "Vector Operations",
};

## Vector Magnitude and Direction

Every vector has two main components: **magnitude (length)** and **direction**.

Consider the example vector <InlineMath math="\overrightarrow{CD}" /> below.

<VectorChart
  title={
    <>
      Vector <InlineMath math="CD" />
    </>
  }
  description={
    <>
      Vector <InlineMath math="CD" /> with a length of 4 cm and a direction of{" "}
      45 degrees relative to the horizontal line.
    </>
  }
  vectors={[
    {
      id: "cd",
      name: "CD",
      points: [
        { x: 0, y: 0 },
        { x: 4 * Math.cos(Math.PI / 4), y: 4 * Math.sin(Math.PI / 4) },
      ],
    },
  ]}
/>

- The **magnitude of vector** <InlineMath math="\overrightarrow{CD}" /> is 4 cm. This indicates the _size_ or _value_ of the vector. The magnitude of a vector <InlineMath math="\vec{v}" /> is usually denoted as <InlineMath math="|\vec{v}|" />. So, <InlineMath math="|\overrightarrow{CD}| = 4" /> cm.
- The **direction of vector** <InlineMath math="\overrightarrow{CD}" /> is 45° relative to the horizontal line. This direction is crucial and distinguishes vectors from scalar quantities (which only have magnitude). Direction can be expressed using angles, compass points (like Northeast), or other references.

## Negative Vector (Opposite Vector)

A negative vector or opposite vector is a vector that has the **same magnitude** but **opposite direction** to the original vector.

Imagine Andi walks 100 m in a direction of 30° (let's call this displacement vector <InlineMath math="\vec{A}" />). Then, Andi returns to the starting position. This second displacement is the opposite vector of <InlineMath math="\vec{A}" />, which we write as <InlineMath math="-\vec{A}" />.

<VectorChart
  title="Vector and Opposite Vector"
  description={
    <>
      Vector <InlineMath math="A" /> and vector <InlineMath math="-A" /> have
      the same magnitude but opposite directions.
    </>
  }
  vectors={[
    {
      id: "A",
      name: "A",
      points: [
        { x: 0, y: 0 },
        { x: 5, y: 2 },
      ],
    },
    {
      id: "-A",
      name: "-A",
      points: [
        { x: 0, y: 0.5 },
        { x: 5, y: 2.5 },
      ],
      direction: "backward",
    },
  ]}
/>

- Vectors <InlineMath math="\vec{A}" /> and <InlineMath math="-\vec{A}" /> have the same magnitude (<InlineMath math="|\vec{A}| = |-\vec{A}|" />).
- The direction of vector <InlineMath math="-\vec{A}" /> is exactly opposite to the direction of vector <InlineMath math="\vec{A}" />. If <InlineMath math="\vec{A}" /> points in one direction, <InlineMath math="-\vec{A}" /> points in the opposite direction (180° difference).

## Zero Vector

The zero vector is a special vector because it has **zero magnitude**. Due to its zero length, this vector **does not have a specific direction**.

The zero vector can be visualized as a single point, where the initial point and terminal point coincide. The zero vector is usually denoted by <InlineMath math="\vec{0}" />.

**Example**:

If Andi walks 100 m east, then walks back 100 m west, Andi's total displacement is zero. This total displacement can be represented as the zero vector (<InlineMath math="\vec{0}" />).

## Equivalent Vectors (Equal Vectors)

Two or more vectors are said to be **equivalent** or **equal** if they have the **same magnitude (length) and direction**, even if their starting points are different.

Consider the graph below showing three equivalent vectors: <InlineMath math="\overrightarrow{CD}" />, <InlineMath math="\overrightarrow{EF}" />, and <InlineMath math="\overrightarrow{KL}" />.

<VectorChart
  title="Equivalent Vectors"
  description={
    <>
      Vectors <InlineMath math="CD" />, <InlineMath math="EF" />, and{" "}
      <InlineMath math="KL" /> have the same magnitude and direction.
    </>
  }
  vectors={[
    {
      id: "CD",
      name: "CD",
      points: [
        { x: 2, y: 1 }, // C
        { x: 5, y: 3 }, // D
      ],
    },
    {
      id: "EF",
      name: "EF",
      points: [
        { x: 4, y: 2 }, // E
        { x: 7, y: 4 }, // F
      ],
    },
    {
      id: "KL",
      name: "KL",
      points: [
        { x: 3, y: 0 }, // K
        { x: 6, y: 2 }, // L
      ],
    },
  ]}
/>

The three vectors above have the same magnitude and direction, so they are equivalent. We can write this as:

<BlockMath math="\overrightarrow{CD} = \overrightarrow{EF} = \overrightarrow{KL}" />

A vector is said to be equivalent to another vector if it has the same magnitude and direction as the other vector.

## Exercise

Consider the two vectors below:

<VectorChart
  title={
    <>
      Comparison of Vector <InlineMath math="A" /> and Vector{" "}
      <InlineMath math="B" />
    </>
  }
  description={
    <>
      Comparison of Vector <InlineMath math="A" /> and Vector{" "}
      <InlineMath math="B" />
    </>
  }
  vectors={[
    {
      id: "A",
      name: "A",
      points: [
        { x: 1, y: 4 },
        { x: 0, y: 0 },
      ],
      direction: "backward",
    },
    {
      id: "B",
      name: "B",
      points: [
        { x: 1, y: 4 },
        { x: 5, y: 5 },
      ],
    },
  ]}
/>

Is vector <InlineMath math="\vec{A}" /> the opposite vector of <InlineMath math="\vec{B}" />?

**Answer:**

Vector <InlineMath math="\vec{A}" /> is **not** the opposite vector of <InlineMath math="\vec{B}" />. To be an opposite vector, two conditions must be met:

1.  **They must have the same magnitude:** Visually, <InlineMath math="|\vec{A}| \neq |\vec{B}|" /> (their lengths appear different).
2.  **Their directions must be exactly opposite (180° apart):** The direction of <InlineMath math="\vec{A}" /> is not opposite to the direction of <InlineMath math="\vec{B}" />.

Since these two conditions are not met, <InlineMath math="\vec{A}" /> is not the opposite vector of <InlineMath math="\vec{B}" />.

**How to make vectors <InlineMath math="\vec{A}" /> and <InlineMath math="\vec{B}" /> opposite?**

To make vectors <InlineMath math="\vec{A}" /> and <InlineMath math="\vec{B}" /> opposite, you could define them like this:

<VectorChart
  title={
    <>
      Opposite Vectors <InlineMath math="A" /> and <InlineMath math="B" />
    </>
  }
  description={
    <>
      Vector <InlineMath math="A" /> and <InlineMath math="B" /> are opposite
    </>
  }
  vectors={[
    {
      id: "A",
      name: "A",
      points: [
        { x: 1, y: 4 },
        { x: 0, y: 0 },
      ],
      direction: "backward",
    },
    {
      id: "B",
      name: "B",
      points: [
        { x: 1, y: 4 },
        { x: 4, y: 8 },
      ],
      direction: "forward",
    },
  ]}
/>
