# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/10/mathematics/vector-operations/vector-addition
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/vector-operations/vector-addition/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 Addition",
  description: "Master vector addition using triangle, parallelogram & polygon methods. Learn resultant calculations, component addition, and real-world applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/12/2025",
  subject: "Vector and Operations",
};

## Basic Concepts of Vector Addition

Vector addition differs from scalar addition. In scalar addition, we only add magnitudes without considering direction. For example, 2 kg of sugar plus 3 kg of sugar equals 5 kg of sugar. However, in vector addition, we must consider both magnitude and direction.

For instance, if someone walks 2 m east and then 3 m west, the result differs from walking 2 m east and then 3 m east again. The result of vector addition is called the **resultant vector**.

## Vector Addition Using the Triangle Method

Vector addition can be performed graphically using the triangle method. The principles are:

1. Draw the first vector <InlineMath math="\vec{a}" />
2. Draw the second vector <InlineMath math="\vec{b}" /> with its tail meeting the tip of the first vector
3. The resultant vector is the vector connecting the tail of the first vector to the tip of the second vector

<div className="mt-4">
  <Vector3d
    title="Vector Addition Using the Triangle Method"
    description="Triangle method for adding vector a and vector b."
    vectors={[
      {
        from: [0, 0, 0],
        to: [4, 1, 0],
        color: getColor("ROSE"),
        label: "vector a",
        labelPosition: "end",
      },
      {
        from: [4, 1, 0],
        to: [6, 4, 0],
        color: getColor("TEAL"),
        label: "vector b",
        labelPosition: "middle",
      },
      {
        from: [0, 0, 0],
        to: [6, 4, 0],
        color: getColor("LIME"),
        label: "vector a+b",
        labelPosition: "end",
      },
    ]}
  />
</div>

Mathematically, if <InlineMath math="\overrightarrow{AB}" /> and <InlineMath math="\overrightarrow{BC}" /> are two vectors, then:

<BlockMath math="\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} = \vec{a} + \vec{b}" />

Vector addition satisfies the commutative property:

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

## Vector Addition Using the Parallelogram Method

Another method for adding vectors is the parallelogram method. The steps are:

1. Draw both vectors with their tails coinciding
2. Create a parallelogram using the two vectors as sides
3. The resultant vector is the diagonal of the parallelogram passing through the common origin of both vectors

<div className="mt-4">
  <Vector3d
    title="Vector Addition Using the Parallelogram Method"
    description="Parallelogram method for adding vector a and vector b."
    vectors={[
      {
        from: [0, 0, 0],
        to: [5, 1, 0],
        color: getColor("ROSE"),
        label: "vector a",
        labelPosition: "end",
      },
      {
        from: [0, 0, 0],
        to: [2, 5, 0],
        color: getColor("TEAL"),
        label: "vector b",
        labelPosition: "middle",
      },
      {
        from: [5, 1, 0],
        to: [7, 6, 0],
        color: getColor("TEAL"),
        label: "vector b",
        labelPosition: "middle",
      },
      {
        from: [2, 5, 0],
        to: [7, 6, 0],
        color: getColor("ROSE"),
        label: "vector a",
        labelPosition: "start",
      },
      {
        from: [0, 0, 0],
        to: [7, 6, 0],
        color: getColor("LIME"),
        label: "a+b = b+a",
        labelPosition: "end",
      },
    ]}
    cameraPosition={[12, 8, 12]}
  />
</div>

This method also satisfies the **commutative property**, so the order of addition does not affect the result.

## Addition Using the Polygon Method

To add more than two vectors, we can use the polygon method. The principle is that the tip of the previous vector meets the tail of the next vector. The resultant vector connects the tail of the first vector to the tip of the last vector.

<Vector3d
  title="Adding Three Vectors Using the Polygon Method"
  description={
    <>
      Addition of vectors <InlineMath math="a + b + c" /> using the polygon
      method.
    </>
  }
  vectors={[
    {
      from: [0, 0, 0],
      to: [3, 1, 0],
      color: getColor("ROSE"),
      label: "vector a",
      labelPosition: "end",
    },
    {
      from: [3, 1, 0],
      to: [5, 4, 0],
      color: getColor("TEAL"),
      label: "vector b",
      labelPosition: "middle",
    },
    {
      from: [5, 4, 0],
      to: [7, 3, 2],
      color: getColor("LIME"),
      label: "vector c",
      labelPosition: "middle",
    },
    {
      from: [0, 0, 0],
      to: [7, 3, 2],
      color: getColor("ORANGE"),
      label: "vector a+b+c",
      labelPosition: "end",
    },
  ]}
  cameraPosition={[12, 8, 12]}
/>

Vector addition using the polygon method also satisfies the associative property:

<BlockMath math="\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}" />

To add three vectors using the parallelogram method, we can:

1. Add two vectors first to get the resultant <InlineMath math="R_1" />
2. Add <InlineMath math="R_1" /> to the third vector to get the final resultant <InlineMath math="R_2" />

<div className="mt-4">
  <Vector3d
    title="Associative Property of Vector Addition"
    description={
      <>
        <InlineMath math="A + (B + C) = (A + B) + C" />
      </>
    }
    vectors={[
      {
        from: [0, 0, 0],
        to: [3, 1, 0],
        color: getColor("ROSE"),
        label: "A",
        labelPosition: "end",
      },
      {
        from: [0, 0, 0],
        to: [1, 3, 0],
        color: getColor("TEAL"),
        label: "B",
        labelPosition: "end",
      },
      {
        from: [0, 0, 0],
        to: [0, 1, 3],
        color: getColor("LIME"),
        label: "C",
        labelPosition: "end",
      },
      {
        from: [1, 3, 0],
        to: [1, 4, 3],
        color: getColor("LIME"),
        label: "C",
        labelPosition: "end",
      },
      {
        from: [0, 0, 0],
        to: [1, 4, 3],
        color: getColor("ORANGE"),
        label: "B+C",
        labelPosition: "middle",
      },
      {
        from: [3, 1, 0],
        to: [4, 5, 3],
        color: getColor("ORANGE"),
        label: "B+C",
        labelPosition: "middle",
      },
      {
        from: [0, 0, 0],
        to: [4, 5, 3],
        color: getColor("PINK"),
        label: "A+(B+C)",
        labelPosition: "end",
      },
    ]}
    cameraPosition={[10, 7, 10]}
  />
</div>

## Vector Addition by Components

Besides graphical methods, vectors can also be added by their components. In a coordinate system, each vector can be expressed in terms of its components.

<Vector3d
  title="Vector Addition by Components"
  description={
    <>
      Vector components on the <InlineMath math="x" />, <InlineMath math="y" />,
      and <InlineMath math="z" /> axes.
    </>
  }
  vectors={[
    {
      from: [0, 0, 0],
      to: [3, 2, 1],
      color: getColor("ROSE"),
      label: "vector a",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [3, 0, 0],
      color: getColor("TEAL"),
      label: "a_x",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [0, 2, 0],
      color: getColor("LIME"),
      label: "a_y",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [0, 0, 1],
      color: getColor("ORANGE"),
      label: "a_z",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [2, 4, 2],
      color: getColor("PINK"),
      label: "vector b",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [5, 6, 3],
      color: getColor("FUCHSIA"),
      label: "a+b",
      labelPosition: "end",
    },
  ]}
  cameraPosition={[9, 8, 9]}
/>

If <InlineMath math="\vec{a} = (a_x, a_y)" /> and <InlineMath math="\vec{b} = (b_x, b_y)" />, then:

<BlockMath math="\vec{a} + \vec{b} = (a_x + b_x, a_y + b_y)" />

Example:

<BlockMath math="\vec{AC} + \vec{AB} = (-3, 3) + (4, 2) = (1, 5)" />

Addition by components gives the same result as graphical addition.

## Applications of Vector Addition in Real Life

One example of vector addition application is in the movement of a boat crossing a river:

1. If the river current velocity is zero (no current), the boat's movement direction remains unchanged, only following the intended direction
2. If the river current velocity is not zero, the boat's movement changes in both direction and speed

<div className="mt-4">
  <Vector3d
    title="Application of Vector Addition: Boat Crossing a River"
    description="Resultant of boat velocity vector and river current vector."
    vectors={[
      {
        from: [0, 0, 0],
        to: [0, 4, 0],
        color: getColor("TEAL"),
        label: "boat velocity",
        labelPosition: "middle",
      },
      {
        from: [0, 0, 0],
        to: [3, 0, 0],
        color: getColor("LIME"),
        label: "river current",
        labelPosition: "end",
      },
      {
        from: [0, 0, 0],
        to: [3, 4, 0],
        color: getColor("ROSE"),
        label: "resultant motion",
        labelPosition: "end",
      },
    ]}
    cameraPosition={[8, 6, 8]}
  />
</div>

The boat's motion is the result of adding the boat's own velocity vector to the river current velocity vector. This is similar to planning vehicle routes, where we need to consider each displacement vector from one location to another to determine the shortest route.

## Difference Between Scalar and Vector Addition

Scalar addition produces only one answer, while vector addition can produce various answers because vectors are related to direction.

**Example of scalar addition**:

3 kg of sugar + 4 kg of sugar = 7 kg of sugar

**Example of vector addition**:

displacement of 3 m east + 4 m east = 7 m east, but displacement of 3 m east + 4 m west = 1 m west.

With the concept of vector addition, we can analyze various physical phenomena involving vector quantities such as displacement, velocity, acceleration, and force.