# Nakafa Framework: LLM

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

Output docs content for large language models.

---

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

export const metadata = {
  title: "Vector Concept",
  description: "Discover what vectors are through real-world examples like wind maps. Learn magnitude, direction, notation methods, and identify valid vector properties.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/08/2025",
  subject: "Vector Operations",
};

## What Is a Vector?

To understand vectors, imagine a weather map showing wind direction and speed at various locations. On such maps, arrows are often drawn. These arrows indicate the **direction** of the wind and **how strong** it blows (usually shown by the length or thickness of the arrow).

In mathematics, quantities that have both **magnitude (size)** and **direction** like this are called **vectors**. Thus, the speed and direction of wind on a weather map is a good example of a vector. The length of the arrow represents the _magnitude_ of the wind speed (for example, 30 km/h), and the direction of the arrow shows the _direction_ in which it blows.

## Drawing and Writing Vectors

Vectors are typically drawn as directed line segments (arrows).

<VectorChart
  title="Vector Graph"
  description={
    <>
      Vectors in a Cartesian coordinate system with horizontal{" "}
      <InlineMath math="x" /> and vertical <InlineMath math="y" />.
    </>
  }
  vectors={[
    {
      id: "vector_1",
      name: "Vector 1",
      points: [
        { x: 0, y: 0 },
        { x: 6, y: 8 },
      ],
    },
    {
      id: "vector_2",
      name: "Vector 2",
      points: [
        { x: 1, y: 1 },
        { x: 5, y: 2 },
      ],
      direction: "backward",
    },
    {
      id: "vector_3",
      name: "Vector 3",
      points: [
        { x: 2, y: 0 },
        { x: 4, y: 7 },
      ],
      direction: "backward",
    },
  ]}
/>

- **Vector 1**: Direction of the vector from point <InlineMath math="(0, 0)" /> to point <InlineMath math="(6, 8)" />
- **Vector 2**: Direction of the vector from point <InlineMath math="(5, 2)" /> to point <InlineMath math="(1, 1)" />
- **Vector 3**: Direction of the vector from point <InlineMath math="(4, 7)" /> to point <InlineMath math="(2, 0)" />

### Key Components of Vectors

1.  **Initial Point:** The starting point of the vector.
2.  **Terminal Point:** The ending point of the vector, marked with an arrowhead.
3.  **Magnitude:** The length of the arrow, representing the value or size of the vector. Often also called _vector length_. The length of vector <InlineMath math="\overrightarrow{AB}" /> is denoted as <InlineMath math="|\overrightarrow{AB}|" />.
4.  **Direction:** The direction pointed by the arrowhead.

### Vector Notation

There are several ways to write vectors:

1.  **Two Capital Letters with an Arrow Above:** Indicating the initial and terminal points. Example: <InlineMath math="\overrightarrow{AB}" />, meaning a vector from point A to point B.
2.  **One Lowercase Letter with an Arrow Above:** Example: <InlineMath math="\vec{a}" />.
3.  **Bold Letters:** Example: **a** or **AB**

## Exercise

Are the shapes below vectors? Why?

<VectorChart
  title="Cartesian Coordinate System"
  description={
    <>
      Horizontal <InlineMath math="x" />
      -axis and vertical <InlineMath math="y" />
      -axis.
    </>
  }
  vectors={[
    {
      id: "shape_1",
      name: "Shape 1",
      points: [
        { x: 0, y: 0 },
        { x: 1, y: 9 },
        { x: 3, y: 4 },
        { x: 6, y: 8 },
      ],
    },
    {
      id: "shape_2",
      name: "Shape 2",
      points: [
        { x: 1, y: 1 },
        { x: 2, y: 6 },
        { x: 5, y: 2 },
      ],
      direction: "backward",
    },
    {
      id: "shape_3",
      name: "Shape 3",
      points: [
        { x: 2, y: 0 },
        { x: 4, y: 7 },
      ],
      direction: "both",
    },
  ]}
/>

These shapes are **not vectors** because:

1. **Shape 1**: because its line is curved, whereas vectors must be represented by straight lines.
2. **Shape 2**: because it consists of more than one straight line, whereas a vector must be represented by a single straight line.
3. **Shape 3**: because it has two directions (shown by arrows at both ends), whereas a vector must have only one clear direction.

A valid vector in mathematics must have these characteristics:

- It must be a straight line
- It must have one direction indicated by an arrow
- It must consist of a single straight line (not broken or curved)
