# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/10/mathematics/vector-operations/opposite-vector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/vector-operations/opposite-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: "Reciprocal Vector",
  description: "Discover reciprocal vectors with inverse magnitude and same direction. Learn formulas, calculations, unit vector relationships, and crystallography applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/12/2025",
  subject: "Vector and Operations",
};

## Understanding Reciprocal Vector

A reciprocal vector is a vector whose magnitude is the reciprocal of the magnitude of the original vector, while maintaining the same direction. If <InlineMath math="\vec{v}" /> is a vector, then the reciprocal vector of <InlineMath math="\vec{v}" /> is a vector with magnitude <InlineMath math="\frac{1}{|\vec{v}|}" />, where <InlineMath math="|\vec{v}|" /> is the magnitude of vector <InlineMath math="\vec{v}" />.

## Formula for Reciprocal Vector

If we have a vector <InlineMath math="\vec{v}" />, then the reciprocal vector of <InlineMath math="\vec{v}" /> can be formulated as:

<BlockMath math="\vec{v}_{\text{reciprocal}} = \frac{1}{|\vec{v}|^2} \cdot \vec{v}" />

Where:

- <InlineMath math="\vec{v}_{\text{reciprocal}}" /> is the reciprocal vector
- <InlineMath math="\vec{v}" /> is the original vector
- <InlineMath math="|\vec{v}|" /> is the magnitude or length of vector <InlineMath math="\vec{v}" />

## Calculation Example

Let's say we have a vector <InlineMath math="\overrightarrow{AB} = 6\mathbf{i} + 8\mathbf{j}" />.

First, we calculate the magnitude of vector <InlineMath math="\overrightarrow{AB}" />:

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

Then, we can find the reciprocal vector of <InlineMath math="\overrightarrow{AB}" />:

<BlockMath math="\overrightarrow{AB}_{\text{reciprocal}} = \frac{1}{|\overrightarrow{AB}|^2} \cdot \overrightarrow{AB} = \frac{1}{10^2} \cdot (6\mathbf{i} + 8\mathbf{j}) = \frac{1}{100} (6\mathbf{i} + 8\mathbf{j})" />

Therefore, the reciprocal vector of <InlineMath math="\overrightarrow{AB}" /> is <InlineMath math="\frac{1}{100}(6\mathbf{i} + 8\mathbf{j})" /> or <InlineMath math="0.06\mathbf{i} + 0.08\mathbf{j}" />.

<Vector3d
  title="Reciprocal Vector Visualization"
  description="Comparison between the original vector (red) and its reciprocal vector (blue)."
  vectors={[
    {
      from: [0, 0, 0],
      to: [6, 8, 0],
      color: getColor("RED"),
      label: "AB",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [0.06, 0.08, 0],
      color: getColor("BLUE"),
      label: "AB reciprocal",
      labelPosition: "end",
    },
  ]}
  cameraPosition={[12, 10, 12]}
/>

## Properties of Reciprocal Vector

1. The magnitude of a reciprocal vector is the reciprocal of the magnitude of the original vector:

   <BlockMath math="|\vec{v}_{\text{reciprocal}}| = \frac{1}{|\vec{v}|}" />

2. The direction of the reciprocal vector is the same as the direction of the original vector.

3. The dot product between a vector and its reciprocal vector:

   <BlockMath math="\vec{v} \cdot \vec{v}_{\text{reciprocal}} = 1" />

4. If <InlineMath math="\vec{v}" /> is a vector with magnitude 1 (unit vector), then its reciprocal vector is the vector itself.

## Applications of Reciprocal Vector

Reciprocal vectors have many applications in physics, mathematics, and engineering. Some important applications include:

1. In crystallography, reciprocal vectors are used to study the lattices of crystalline solids.

2. In quantum physics, reciprocal vectors are used in momentum space calculations.

3. In vector analysis, reciprocal vectors help solve vector equations.

## Unit Vector and Its Relationship with Reciprocal Vector

### Unit Vector

A unit vector is a vector with a magnitude of exactly one unit. The unit vector of a vector <InlineMath math="\vec{v}" /> is usually denoted by <InlineMath math="\hat{v}" /> and is calculated using the formula:

<BlockMath math="\hat{v} = \frac{\vec{v}}{|\vec{v}|}" />

<div className="mt-4">
  <Vector3d
    title="Comparison of Original Vector and Unit Vector"
    description="Original vector and its corresponding unit vector."
    vectors={[
      {
        from: [0, 0, 0],
        to: [6, 8, 0],
        color: getColor("GREEN"),
        label: "v",
        labelPosition: "end",
      },
      {
        from: [0, 0, 0],
        to: [0.6, 0.8, 0],
        color: getColor("VIOLET"),
        label: "v̂ (unit)",
        labelPosition: "end",
      },
    ]}
    cameraPosition={[12, 10, 12]}
  />
</div>

### Relationship with Reciprocal Vector

There is an interesting relationship between unit vectors and reciprocal vectors. If <InlineMath math="\vec{v}" /> is a vector, then:

<BlockMath math="\vec{v}_{\text{reciprocal}} = \frac{\hat{v}}{|\vec{v}|}" />

This shows that a reciprocal vector can be obtained by dividing the unit vector by the magnitude of the original vector.