# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/10/mathematics/vector-operations/equivalent-vector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/vector-operations/equivalent-vector/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: "Equivalent Vectors",
  description: "Master equivalent vectors with same magnitude and direction. Learn properties, component representation, and real-world applications in physics and engineering.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/12/2025",
  subject: "Vector and Operations",
};

## Definition of Equivalent Vectors

Two vectors are said to be equivalent if they have the same magnitude (length) and direction. Mathematically, two vectors <InlineMath math="\vec{a}" /> and <InlineMath math="\vec{b}" /> are equivalent if their components are equal. In mathematical notation, this can be written as <InlineMath math="\vec{a} \equiv \vec{b}" />.

Equivalent vectors can have different positions in a plane or space, but they maintain the same magnitude and direction.

<Vector3d
  title="Example of Equivalent Vectors"
  description="The purple and orange vectors have the same magnitude and direction, making them equivalent despite their different positions."
  vectors={[
    {
      from: [0, 0, 0],
      to: [3, 2, 1],
      color: getColor("VIOLET"),
      label: "v₁",
    },
    {
      from: [1, 1, 1],
      to: [4, 3, 2],
      color: getColor("ORANGE"),
      label: "v₂",
    },
  ]}
  cameraPosition={[8, 5, 8]}
/>

## Conditions for Equivalent Vectors

Two vectors <InlineMath math="\overrightarrow{CD}" /> and <InlineMath math="\overrightarrow{PQ}" /> are said to be equivalent if:

1. Both vectors have equal length: <InlineMath math="|\overrightarrow{CD}| = |\overrightarrow{PQ}|" />
2. Both vectors have the same direction

## Representation of Equivalent Vectors

### In Component Form

In a two-dimensional Cartesian plane, two vectors <InlineMath math="\vec{a}" /> and <InlineMath math="\vec{b}" /> are equivalent if:

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

where <InlineMath math="a_1 = b_1" /> and <InlineMath math="a_2 = b_2" />

In three-dimensional space, vectors <InlineMath math="\vec{a}" /> and <InlineMath math="\vec{b}" /> are equivalent if:

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

where <InlineMath math="a_1 = b_1" />, <InlineMath math="a_2 = b_2" />, and <InlineMath math="a_3 = b_3" />

<Vector3d
  title="Equivalent Vectors in Component Form"
  description="Two vectors with the same components are always equivalent, even if they have different positions in space."
  vectors={[
    {
      from: [0, 0, 0],
      to: [2, 3, 1],
      color: getColor("FUCHSIA"),
      label: "a = (2, 3, 1)",
    },
    {
      from: [3, 0, 2],
      to: [5, 3, 3],
      color: getColor("CYAN"),
      label: "b = (2, 3, 1)",
    },
  ]}
/>

### In Terms of Initial and Terminal Points

If vector <InlineMath math="\overrightarrow{AB}" /> has initial point <InlineMath math="A(x_1, y_1)" /> and terminal point <InlineMath math="B(x_2, y_2)" />, then the vector can be expressed as:

<BlockMath math="\overrightarrow{AB} = (x_2 - x_1, y_2 - y_1)" />

Two vectors <InlineMath math="\overrightarrow{AB}" /> and <InlineMath math="\overrightarrow{CD}" /> are equivalent if:

<BlockMath math="(x_2 - x_1, y_2 - y_1) = (x_4 - x_3, y_4 - y_3)" />

where <InlineMath math="C(x_3, y_3)" /> and <InlineMath math="D(x_4, y_4)" />

## Properties of Equivalent Vectors

### Reflexive Property

Every vector is equivalent to itself.

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

### Symmetric Property

If vector <InlineMath math="\vec{a}" /> is equivalent to vector <InlineMath math="\vec{b}" />, then vector <InlineMath math="\vec{b}" /> is also equivalent to vector <InlineMath math="\vec{a}" />.

<BlockMath math="\text{If } \vec{a} \equiv \vec{b} \text{, then } \vec{b} \equiv \vec{a}" />

### Transitive Property

If vector <InlineMath math="\vec{a}" /> is equivalent to vector <InlineMath math="\vec{b}" /> and vector <InlineMath math="\vec{b}" /> is equivalent to vector <InlineMath math="\vec{c}" />, then vector <InlineMath math="\vec{a}" /> is equivalent to vector <InlineMath math="\vec{c}" />.

<div className="flex flex-col gap-4">
  <BlockMath math="\text{If } \vec{a} \equiv \vec{b} \text{ and } \vec{b} \equiv \vec{c} \text{, then } \vec{a} \equiv \vec{c}" />

  <Vector3d
    title="Transitive Property of Equivalent Vectors"
    description={
      <>
        Three equivalent vectors: if <InlineMath math="a \equiv b" /> and{" "}
        <InlineMath math="b \equiv c" />, then <InlineMath math="a \equiv c" />.
      </>
    }
    vectors={[
      {
        from: [0, 0, 0],
        to: [2, 2, 0],
        color: getColor("AMBER"),
        label: "a",
      },
      {
        from: [1, 1, 2],
        to: [3, 3, 2],
        color: getColor("EMERALD"),
        label: "b",
      },
      {
        from: [2, 0, 1],
        to: [4, 2, 1],
        color: getColor("PINK"),
        label: "c",
      },
    ]}
    cameraPosition={[7, 5, 7]}
  />
</div>

## Examples of Equivalent Vectors

### Example 1

Vector <InlineMath math="\overrightarrow{AB}" /> with <InlineMath math="A(2, 3)" /> and <InlineMath math="B(5, 7)" /> is equivalent to vector <InlineMath math="\overrightarrow{CD}" /> with <InlineMath math="C(1, 1)" /> and <InlineMath math="D(4, 5)" />.

Proof:

<div className="flex flex-col gap-4">
  <BlockMath math="\overrightarrow{AB} = (5-2, 7-3) = (3, 4)" />
  <BlockMath math="\overrightarrow{CD} = (4-1, 5-1) = (3, 4)" />
</div>

Since <InlineMath math="\overrightarrow{AB} = \overrightarrow{CD} = (3, 4)" />, vector <InlineMath math="\overrightarrow{AB}" /> is equivalent to vector <InlineMath math="\overrightarrow{CD}" />.

<Vector3d
  title="Vectors AB and CD"
  description={
    <>
      Visualization of vectors <InlineMath math="AB(3,4,0)" /> and{" "}
      <InlineMath math="CD(3,4,0)" /> which are equivalent in space.
    </>
  }
  vectors={[
    {
      from: [2, 3, 0],
      to: [5, 7, 0],
      color: getColor("VIOLET"),
      label: "AB",
    },
    {
      from: [1, 1, 0],
      to: [4, 5, 0],
      color: getColor("YELLOW"),
      label: "CD",
    },
  ]}
  cameraPosition={[10, 10, 10]}
/>

### Example 2

Vector <InlineMath math="\overrightarrow{PQ}" /> with <InlineMath math="P(0, 0)" /> and <InlineMath math="Q(2, 2)" /> is equivalent to vector <InlineMath math="\overrightarrow{RS}" /> with <InlineMath math="R(3, 1)" /> and <InlineMath math="S(5, 3)" />.

Proof:

<div className="flex flex-col gap-4">
  <BlockMath math="\overrightarrow{PQ} = (2-0, 2-0) = (2, 2)" />
  <BlockMath math="\overrightarrow{RS} = (5-3, 3-1) = (2, 2)" />
</div>

Since <InlineMath math="\overrightarrow{PQ} = \overrightarrow{RS} = (2, 2)" />, vector <InlineMath math="\overrightarrow{PQ}" /> is equivalent to vector <InlineMath math="\overrightarrow{RS}" />.

<Vector3d
  title="Vectors PQ and RS"
  description={
    <>
      Visualization of vectors <InlineMath math="PQ(2,2,0)" /> and{" "}
      <InlineMath math="RS(2,2,0)" /> which are equivalent in space.
    </>
  }
  vectors={[
    {
      from: [0, 0, 0],
      to: [2, 2, 0],
      color: getColor("TEAL"),
      label: "PQ",
    },
    {
      from: [3, 1, 0],
      to: [5, 3, 0],
      color: getColor("ROSE"),
      label: "RS",
    },
  ]}
  cameraPosition={[8, 6, 8]}
/>

## Applications of Equivalent Vectors

The concept of equivalent vectors is important in various applications, including:

1. In physics, for calculating displacement, velocity, and acceleration of objects
2. In navigation, for determining direction and travel distance
3. In computer graphics, for object transformation
4. In electrical engineering, for representing magnetic and electric forces