# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/vector-operations/vector-subtraction
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/vector-operations/vector-subtraction/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 Subtraction",
  description: "Master vector subtraction with step-by-step examples, geometric visualization, and real-world applications. Learn algebraic methods and properties.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/12/2025",
  subject: "Vector and Operations",
};

## Basic Concept of Vector Subtraction

Vector subtraction is one of the basic operations in vector mathematics. Unlike vector addition that combines two vectors, vector subtraction produces the difference between two vectors. When we subtract vector <InlineMath math="\vec{b}" /> from vector <InlineMath math="\vec{a}" />, the result is a vector that, when added to <InlineMath math="\vec{b}" />, will produce <InlineMath math="\vec{a}" />.

Mathematically, vector subtraction is defined as:

<BlockMath math="\vec{a} - \vec{b} = \vec{a} + (-\vec{b})" />

This means that subtracting vector <InlineMath math="\vec{b}" /> from vector <InlineMath math="\vec{a}" /> is equivalent to adding vector <InlineMath math="\vec{a}" /> with the negative of vector <InlineMath math="\vec{b}" />.

## Geometric Vector Subtraction

Geometrically, vector subtraction <InlineMath math="\vec{a} - \vec{b}" /> can be depicted by:

1. Drawing vectors <InlineMath math="\vec{a}" /> and <InlineMath math="\vec{b}" /> with the same initial point.
2. Drawing vector <InlineMath math="-\vec{b}" /> (vector <InlineMath math="\vec{b}" /> with reversed direction).
3. Drawing a vector from the endpoint of <InlineMath math="\vec{b}" /> to the endpoint of <InlineMath math="\vec{a}" />.

The resulting vector, <InlineMath math="\vec{a} - \vec{b}" />, can also be obtained by drawing a line from the endpoint of vector <InlineMath math="\vec{b}" /> to the endpoint of vector <InlineMath math="\vec{a}" /> when both vectors are drawn from the same origin point.

<Vector3d
  title="Visualization of Vector Subtraction in 3D Space"
  description={
    <>
      Vector subtraction <InlineMath math="a - b" /> is equivalent to vector{" "}
      <InlineMath math="a" /> added to the negative of vector{" "}
      <InlineMath math="b" />.
    </>
  }
  vectors={[
    {
      from: [0, 0, 0],
      to: [3, 2, 1],
      color: getColor("TEAL"),
      label: "a",
    },
    {
      from: [0, 0, 0],
      to: [1, 2, 2],
      color: getColor("ROSE"),
      label: "b",
    },
    {
      from: [0, 0, 0],
      to: [-1, -2, -2],
      color: getColor("ORANGE"),
      label: "-b",
    },
    {
      from: [0, 0, 0],
      to: [2, 0, -1],
      color: getColor("PINK"),
      label: "a - b",
    },
  ]}
  cameraPosition={[6, 4, 6]}
/>

## Algebraic Vector Subtraction

Vector subtraction can be performed by subtracting corresponding components. Suppose we have two vectors:

<div className="flex flex-col gap-4">
  <BlockMath math="\vec{a} = (a_1, a_2, a_3)" />
  <BlockMath math="\vec{b} = (b_1, b_2, b_3)" />
</div>

Then vector subtraction <InlineMath math="\vec{a} - \vec{b}" /> can be calculated as:

<BlockMath math="\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3)" />

For two-dimensional vectors, the equation becomes:

<BlockMath math="\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2)" />

## Example of Vector Subtraction Calculation

Suppose there are two vectors:

<div className="flex flex-col gap-4">
  <BlockMath math="\vec{a} = (3, 4)" />
  <BlockMath math="\vec{b} = (1, 2)" />
</div>

Vector subtraction <InlineMath math="\vec{a} - \vec{b}" /> is:

<BlockMath math="\vec{a} - \vec{b} = (3, 4) - (1, 2) = (3-1, 4-2) = (2, 2)" />

While vector subtraction <InlineMath math="\vec{b} - \vec{a}" /> is:

<BlockMath math="\vec{b} - \vec{a} = (1, 2) - (3, 4) = (1-3, 2-4) = (-2, -2)" />

Note that <InlineMath math="\vec{a} - \vec{b} \neq \vec{b} - \vec{a}" /> in general. In fact, <InlineMath math="\vec{a} - \vec{b} = -(\vec{b} - \vec{a})" />.

<Vector3d
  title="Example of Vector Subtraction"
  description={
    <>
      Visualization of vector subtraction{" "}
      <InlineMath math="a - b = (3,4,0) - (1,2,0) = (2,2,0)" />.
    </>
  }
  vectors={[
    {
      from: [0, 0, 0],
      to: [3, 4, 0],
      color: getColor("TEAL"),
      label: "a",
    },
    {
      from: [0, 0, 0],
      to: [1, 2, 0],
      color: getColor("ROSE"),
      label: "b",
    },
    {
      from: [0, 0, 0],
      to: [2, 2, 0],
      color: getColor("LIME"),
      label: "a - b",
    },
  ]}
  cameraPosition={[8, 6, 8]}
/>

## Applications of Vector Subtraction

Vector subtraction has many applications in real life:

1. **Calculating Displacement**: If <InlineMath math="\vec{a}" /> is the final position and <InlineMath math="\vec{b}" /> is the initial position, then <InlineMath math="\vec{a} - \vec{b}" /> is the displacement vector.

2. **Calculating Distance**: In games like Angry Birds, vector subtraction is used to calculate the distance and direction between the bird and the target.

3. **Physics**: Vector subtraction is used to calculate the resultant force in systems with multiple forces.

## Vector Subtraction to Find the Resultant

Vector subtraction can also be used to find the resultant vector. A resultant vector is a vector that represents the combined effect of two or more vectors.

Suppose an object moves from initial position <InlineMath math="\vec{p}_1" /> to final position <InlineMath math="\vec{p}_2" />. The displacement vector of the object is:

<BlockMath math="\vec{d} = \vec{p}_2 - \vec{p}_1" />

This resultant vector shows the direction and distance of the object's displacement.

## Properties of Vector Subtraction

Vector subtraction has several important properties:

1. <InlineMath math="\vec{a} - \vec{b} = \vec{a} + (-\vec{b})" />
2. <InlineMath math="\vec{a} - \vec{a} = \vec{0}" /> (zero vector)
3. <InlineMath math="\vec{a} - \vec{0} = \vec{a}" />
4. <InlineMath math="\vec{0} - \vec{a} = -\vec{a}" />
5. <InlineMath math="\vec{a} - \vec{b} = -(\vec{b} - \vec{a})" />

## Example Problem

A monkey is at position (3, 4) and wants to get a banana located at position (2, 1). Determine the displacement vector of the monkey to reach the banana.

**Solution:**

Monkey's position: <InlineMath math="\vec{m} = (3, 4)" />
Banana's position: <InlineMath math="\vec{p} = (2, 1)" />

Displacement vector of the monkey to the banana:

<BlockMath math="\vec{d} = \vec{p} - \vec{m} = (2, 1) - (3, 4) = (2-3, 1-4) = (-1, -3)" />

Therefore, the monkey needs to move 1 unit in the negative x-axis direction and 3 units in the negative y-axis direction to reach the banana.

The magnitude of the displacement vector can be calculated using the Pythagorean theorem:

<BlockMath math="|\vec{d}| = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \approx 3.16" />