# Nakafa Framework: LLM

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

Output docs content for large language models.

---

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

export const metadata = {
  title: "Two-Dimensional Vector",
  description: "Master two-dimensional vectors on coordinate planes. Learn vector components, magnitude calculations, and unit vectors with visual demonstrations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/11/2025",
  subject: "Vector and Operations",
};

## Two-Dimensional Vector Concept

In the Cartesian coordinate system, each point on a plane can be represented by a pair of numbers <InlineMath math="(x, y)" />, where <InlineMath math="x" /> is the horizontal position and <InlineMath math="y" /> is the vertical position. The origin point is <InlineMath math="O(0, 0)" />.

If we draw a straight line from the origin <InlineMath math="O" /> to another point, for example <InlineMath math="Q(x, y)" />, we get a **vector**. This vector is often written as <InlineMath math="\overrightarrow{OQ}" />. A vector has both **magnitude** (line length) and **direction** (indicated by the arrow).

To simplify, we use **unit vectors**. Unit vectors have a length of 1 unit.

- <InlineMath math="\mathbf{i}" /> is the unit vector in the positive <InlineMath math="x" />
  -axis direction (horizontal).
- <InlineMath math="\mathbf{j}" /> is the unit vector in the positive <InlineMath math="y" />
  -axis direction (vertical).

Vector <InlineMath math="\overrightarrow{OQ}" /> can be expressed as a combination of horizontal movement of <InlineMath math="x" /> and vertical movement of <InlineMath math="y" />. In unit vector form, we write:

<BlockMath math="\overrightarrow{OQ} = x\mathbf{i} + y\mathbf{j}" />

## Vector Components and Magnitude

The values <InlineMath math="x" /> and <InlineMath math="y" /> in vector <InlineMath math="\overrightarrow{OQ} = x\mathbf{i} + y\mathbf{j}" /> are called the vector **components**.

- <InlineMath math="x" /> is the **horizontal component**. It's like the shadow of
  the vector on the <InlineMath math="x" />
  -axis when illuminated from above.
- <InlineMath math="y" /> is the **vertical component**. It's like the shadow of
  the vector on the <InlineMath math="y" />
  -axis when illuminated from the side.

A vector with these two components is called a **two-dimensional vector**.

The **length** or **magnitude** of vector <InlineMath math="\overrightarrow{OQ}" />, written as <InlineMath math="|\overrightarrow{OQ}|" />, is the distance from the origin point <InlineMath math="O" /> to the endpoint <InlineMath math="Q" />. If <InlineMath math="Q(x,y)" /> is the endpoint of the vector and <InlineMath math="R(x,0)" /> is the projection of point Q onto the <InlineMath math="x" />-axis, we can calculate it using the Pythagorean theorem on the right-angled triangle <InlineMath math="ORQ" />:

<BlockMath math="|\overrightarrow{OQ}| = \sqrt{x^2 + y^2}" />

## Two-Dimensional Vector Visualization

Observe the following vector visualization to understand the concept of vectors in the Cartesian plane:

<Vector3d
  title="Vector in the Coordinate Plane"
  description={
    <>
      Visualization of vector{" "}
      <InlineMath math="OQ = 4\mathbf{i} + 3\mathbf{j}" />
      in the Cartesian coordinate system
    </>
  }
  vectors={[
    {
      from: [0, 0, 0],
      to: [4, 3, 0],
      color: getColor("LIME"),
      label: "Vector OQ",
    },
    {
      from: [0, 0, 0],
      to: [4, 0, 0],
      color: getColor("TEAL"),
      label: "X Component",
    },
    {
      from: [0, 0, 0],
      to: [0, 3, 0],
      color: getColor("YELLOW"),
      label: "Y Component",
    },
  ]}
/>

In this visualization, we don't use the <InlineMath math="z" />-axis because we're working in a two-dimensional plane.

In the visualization above:

- Vector <InlineMath math="\overrightarrow{OQ}" /> (light green) has initial point <InlineMath math="O(0,0)" /> and endpoint <InlineMath math="Q(4,3)" />
- The <InlineMath math="x" />-component (light blue) is the projection of vector <InlineMath math="\overrightarrow{OQ}" /> on the <InlineMath math="x" />-axis, which is <InlineMath math="4\mathbf{i}" />
- The <InlineMath math="y" />-component (yellow) is the projection of vector <InlineMath math="\overrightarrow{OQ}" /> on the <InlineMath math="y" />-axis, which is <InlineMath math="3\mathbf{j}" />
- The magnitude of the vector <InlineMath math="|\overrightarrow{OQ}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5" />