# Nakafa Framework: LLM

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

Output docs content for large language models.

---

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

export const metadata = {
  title: "Vector Components",
  description: "Learn to break down vectors into x, y, z components, calculate magnitude using Pythagorean theorem, and find unit direction vectors with examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/11/2025",
  subject: "Vector and Operations",
};

## Understanding Vector Components

In studying vectors, we need to understand that every vector can be broken down into its components. Vector components are parts of a vector that are parallel to the coordinate axes.

Vector components are values that indicate how far a vector moves in the direction of the x-axis and y-axis. Every vector in a plane can be expressed as a linear combination of unit vectors <InlineMath math="i" /> and <InlineMath math="j" />.

<Vector3d
  title="Vector Visualization and Its Components"
  description={
    <>
      Vector <InlineMath math="AB" /> and its components on the{" "}
      <InlineMath math="x" />, <InlineMath math="y" />, and{" "}
      <InlineMath math="z" /> axes.
    </>
  }
  vectors={[
    {
      from: [0, 0, 0],
      to: [4, 6, 0],
      color: getColor("ROSE"),
      label: "AB",
    },
    {
      from: [0, 0, 0],
      to: [4, 0, 0],
      color: getColor("TEAL"),
      label: "x component",
    },
    {
      from: [0, 0, 0],
      to: [0, 6, 0],
      color: getColor("LIME"),
      label: "y component",
    },
  ]}
/>

If we have a vector <InlineMath math="\overrightarrow{AB}" />, then:

<BlockMath math="\overrightarrow{AB} = a\cdot i + b\cdot j" />

where:

- <InlineMath math="a" /> is the vector component on the x-axis (horizontal)
- <InlineMath math="b" /> is the vector component on the y-axis (vertical)
- <InlineMath math="i" /> is the unit vector in the direction of the x-axis
- <InlineMath math="j" /> is the unit vector in the direction of the y-axis

### Example of Vector Components

Consider the vector <InlineMath math="\overrightarrow{AB}" /> in the figure. This vector can be written as:

<BlockMath math="\overrightarrow{AB} = 6i + 8j" />

This means that vector <InlineMath math="\overrightarrow{AB}" /> has a horizontal component of 6 units to the right and a vertical component of 8 units upward.

## Vector Magnitude from Its Components

When we know the components of a vector, we can calculate the length or magnitude of the vector using the Pythagorean theorem.

The magnitude of vector <InlineMath math="\overrightarrow{AB}" /> is denoted by <InlineMath math="|\overrightarrow{AB}|" /> and calculated using the formula:

<BlockMath math="|\overrightarrow{AB}| = \sqrt{a^2 + b^2}" />

where <InlineMath math="a" /> and <InlineMath math="b" /> are the components of the vector.

### Example of Vector Magnitude Calculation

For the vector <InlineMath math="\overrightarrow{AB} = 6i + 8j" />, its magnitude is:

<BlockMath math="|\overrightarrow{AB}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10" />

Therefore, the magnitude of vector <InlineMath math="\overrightarrow{AB}" /> is 10 units.

## Vectors in Three-Dimensional Space

Vectors are not limited to a plane (two dimensions) but can also be extended to three-dimensional space.

<Vector3d
  title="Vectors in 3D Space"
  description="Visualization of a vector and its components in three-dimensional space."
  vectors={[
    {
      from: [0, 0, 0],
      to: [5, 7, 4],
      color: getColor("ORANGE"),
      label: "3D Vector",
    },
    {
      from: [0, 0, 0],
      to: [5, 0, 0],
      color: getColor("TEAL"),
      label: "x component",
    },
    {
      from: [0, 0, 0],
      to: [0, 7, 0],
      color: getColor("LIME"),
      label: "y component",
    },
    {
      from: [0, 0, 0],
      to: [0, 0, 4],
      color: getColor("YELLOW"),
      label: "z component",
    },
  ]}
  cameraPosition={[12, 8, 12]}
/>

In three-dimensional space, a vector has three components: the x-component, y-component, and z-component. A vector in three-dimensional space can be expressed as:

<BlockMath math="\overrightarrow{v} = ai + bj + ck" />

where:

- <InlineMath math="a" /> is the vector component on the x-axis
- <InlineMath math="b" /> is the vector component on the y-axis
- <InlineMath math="c" /> is the vector component on the z-axis
- <InlineMath math="i" />, <InlineMath math="j" />, and <InlineMath math="k" /> are
  unit vectors in the direction of the x, y, and z axes

The magnitude of a vector in three-dimensional space is calculated using the formula:

<BlockMath math="|\overrightarrow{v}| = \sqrt{a^2 + b^2 + c^2}" />

## Unit Direction Vector

To determine the direction of a vector, we can use a unit vector. A unit vector is a vector with a magnitude of 1 unit. To obtain a unit vector from a vector, we divide the vector by its magnitude.

The unit direction vector of <InlineMath math="\overrightarrow{AB}" /> is denoted by <InlineMath math="\hat{AB}" /> and calculated using:

<BlockMath math="\hat{AB} = \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|}" />

### Example of Unit Direction Vector

For the vector <InlineMath math="\overrightarrow{AB} = 6i + 8j" /> with a magnitude of 10, its unit direction vector is:

<BlockMath math="\hat{AB} = \frac{6i + 8j}{10} = \frac{6}{10}i + \frac{8}{10}j = 0.6i + 0.8j" />

This unit vector indicates the direction of vector <InlineMath math="\overrightarrow{AB}" /> without regard to its magnitude.

## Applications of Vector Components

Vector components have many applications in everyday life, such as:

- Calculating velocity and displacement in physics
- Analyzing forces in mechanics
- Determining the direction and magnitude of resultants in object movement
- Navigation and position determination in coordinate systems

By understanding vector components, we can analyze various problems involving direction and magnitude in mathematics and other applied sciences.