# Nakafa Framework: LLM

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

Output docs content for large language models.

---

import { Vector3d } from "@repo/design-system/components/contents/vector-3d";
import { VectorChart } from "@repo/design-system/components/contents/vector-chart";
import { getColor } from "@repo/design-system/lib/color";

export const metadata = {
  title: "Vectors and Coordinate System",
  description: "Master Cartesian coordinates in 2D and 3D space. Learn position vectors, displacement vectors, notation methods, and real-world GPS applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/09/2025",
  subject: "Vectors and Operations",
};

## Cartesian Coordinate System

To specify the position of an object, such as an aircraft in flight, we need a **frame of reference** or **coordinate system**. The most commonly used system is the **Cartesian Coordinate System**, developed by René Descartes.

The Cartesian Coordinate System uses perpendicular axes to determine the location of a point.

- In **two dimensions (2D)**, we use the <InlineMath math="x" /> axis (horizontal) and <InlineMath math="y" /> axis (vertical). A point's position is expressed as an ordered pair <InlineMath math="(x, y)" />, for example, point <InlineMath math="P(3, 4)" />.

  This means that point <InlineMath math="P" /> is located <InlineMath math="3" /> units to the right (positive <InlineMath math="x" /> axis) and <InlineMath math="4" /> units upward (positive <InlineMath math="y" /> axis) from the origin <InlineMath math="O(0, 0)" />.

  <VectorChart
    title={
      <>
        Position Vector <InlineMath math="P(3, 4)" />
      </>
    }
    description={
      <>
        Origin <InlineMath math="O(0, 0)" /> to point{" "}
        <InlineMath math="P(3, 4)" />.
      </>
    }
    vectors={[
      {
        id: "OP",
        name: "OP",
        points: [
          { x: 0, y: 0 },
          { x: 3, y: 4 },
        ],
      },
    ]}
  />

- In **three dimensions (3D)**, we add a <InlineMath math="z" /> axis that is perpendicular to the <InlineMath math="xy" /> plane. A point's position is expressed as <InlineMath math="(x, y, z)" />, used for determining the position of objects like aircraft in space.

  We can think of <InlineMath math="x" /> as length, <InlineMath math="y" /> as width, and <InlineMath math="z" /> as height.

  <Vector3d
    title={
      <>
        Position Vector <InlineMath math="P(3, 4, 5)" />
      </>
    }
    description={
      <>
        Origin <InlineMath math="O(0, 0, 0)" /> to point{" "}
        <InlineMath math="P(3, 4, 5)" />.
      </>
    }
    vectors={[
      {
        from: [0, 0, 0],
        to: [3, 4, 5],
        color: getColor("PURPLE"),
        label: "P(3, 4, 5)",
      },
    ]}
  />

## Difference Between Vectors and Coordinates

Unlike coordinates which represent the _position_ of a point, a **vector** is a quantity that has both **magnitude (length)** and **direction**. Imagine a fly traveling from point <InlineMath math="P(3, 4)" /> to point <InlineMath math="Q(-2, -8)" />. The fly's displacement can be represented as a vector.

- **Position Vector:** A vector starting from the origin <InlineMath math="O(0, 0)" /> to a point <InlineMath math="P(x, y)" />. The position vector of <InlineMath math="P" /> is written as <InlineMath math="\overrightarrow{OP}" />.

  Example: <InlineMath math="\overrightarrow{OP} = (3, 4)" />.

- **Displacement Vector (or Free Vector):** A vector connecting any two points, for example, from point <InlineMath math="P(x_P, y_P)" /> to point <InlineMath math="Q(x_Q, y_Q)" />. This vector is written as <InlineMath math="\overrightarrow{PQ}" />. It doesn't have to start from the origin.

  Example: displacement vectors from <InlineMath math="P(3, 4)" /> to <InlineMath math="Q(-2, -8)" /> or from <InlineMath math="Q(-2, -8)" /> to <InlineMath math="R(1, 3)" />.

  <VectorChart
    title="Displacement Vectors"
    description={
      <>
        Displacement vectors from <InlineMath math="P" /> to{" "}
        <InlineMath math="Q" /> or from <InlineMath math="Q" /> to{" "}
        <InlineMath math="R" />
      </>
    }
    vectors={[
      {
        id: "QR",
        name: "QR",
        points: [
          { x: -2, y: -8 },
          { x: 1, y: 3 },
        ],
      },
      {
        id: "PQ",
        name: "PQ",
        points: [
          { x: 3, y: 4 },
          { x: -2, y: -8 },
        ],
        direction: "backward",
      },
    ]}
  />

## Vector Notation

Vectors can be written in several ways:

- **Row Vector:** Using regular parentheses.

  Example: <InlineMath math="\overrightarrow{OP} = (3, 4)" />.

- **Column Vector:** Using square brackets or large parentheses.

  Example: <InlineMath math="\overrightarrow{OP} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}" />.

- **Unit Vector Notation (for 3D):** Using a combination of unit vectors <InlineMath math="\mathbf{i}" />, <InlineMath math="\mathbf{j}" />, and <InlineMath math="\mathbf{k}" /> which are aligned with the <InlineMath math="x" />, <InlineMath math="y" />, and <InlineMath math="z" /> axes.

  Example: a vector <InlineMath math="\overrightarrow{AB}" /> can be written as <InlineMath math="\mathbf{i} - 3\mathbf{j} - 4\mathbf{k}" />, meaning vector <InlineMath math="\overrightarrow{AB}" /> has a component of <InlineMath math="1" /> in the <InlineMath math="x" /> direction, <InlineMath math="-3" /> in the <InlineMath math="y" /> direction, and <InlineMath math="-4" /> in the <InlineMath math="z" /> direction.

## Why Are Coordinate Systems Important?

Imagine a map application on your phone. This application relies on coordinate systems to:

1.  Determine your current location.
2.  Find your destination.
3.  Calculate the shortest or fastest route.
4.  Provide direction instructions (turn left, turn right, go straight).

Without coordinate systems, it would be impossible for applications to accurately understand position and direction. Every movement, such as an aircraft changing altitude or direction, needs to be reported and processed using coordinate systems and vector concepts.