# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/vector-operations/position-vector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/vector-operations/position-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: "Position Vector",
  description: "Master position vectors from origin to points. Learn coordinate representation, displacement relationships, GPS applications, and 3D space visualization.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/12/2025",
  subject: "Vector and Operations",
};

## Definition of Position Vector

A position vector is a vector that starts from point O (origin) in a coordinate system and ends at another point. This vector plays an important role in determining the position or location of a point in a coordinate system.

<Vector3d
  title="Position Vector Visualization"
  description="Examples of position vectors from origin O to points A and B."
  vectors={[
    {
      from: [0, 0, 0],
      to: [-3, 2, 0],
      color: getColor("PINK"),
      label: "OA",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [7, 5, 0],
      color: getColor("VIOLET"),
      label: "OB",
      labelPosition: "end",
    },
  ]}
  cameraPosition={[10, 6, 10]}
/>

### Characteristics of Position Vectors

Each position vector has the following characteristics:

- Always starts from the origin O (center of coordinates)
- Ends at a specific point in the coordinate system
- The coordinates of the position vector are the same as the coordinates of its endpoint

## Representation of Position Vectors

In general, if we have a point P with coordinates <InlineMath math="(x, y)" /> in a plane, then the position vector from point O to point P can be written as <InlineMath math="\overrightarrow{OP} = (x, y)" />.

In three-dimensional space, if point P has coordinates <InlineMath math="(x, y, z)" />, then its position vector is <InlineMath math="\overrightarrow{OP} = (x, y, z)" />.

In the visualization below, we use the notation OA, OB, OC, and OD to indicate position vectors from point O to specific points (A, B, C, or D).

<Vector3d
  title="Position Vectors in 3D Space"
  description="Examples of several position vectors in three-dimensional space."
  vectors={[
    {
      from: [0, 0, 0],
      to: [4, 0, 0],
      color: getColor("ROSE"),
      label: "OA",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [0, 4, 0],
      color: getColor("GREEN"),
      label: "OB",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [0, 0, 4],
      color: getColor("BLUE"),
      label: "OC",
      labelPosition: "end",
    },
    {
      from: [0, 0, 0],
      to: [3, 3, 3],
      color: getColor("YELLOW"),
      label: "OD",
      labelPosition: "end",
    },
  ]}
  cameraPosition={[6, 8, 8]}
/>

## Examples of Position Vectors

Suppose there are two points A and B in the coordinate plane:

- Point A with coordinates <InlineMath math="(-3, 2)" />
- Point B with coordinates <InlineMath math="(7, 5)" />

Then the position vectors of these two points are:

- <InlineMath math="\overrightarrow{OA} = (-3, 2)" />
- <InlineMath math="\overrightarrow{OB} = (7, 5)" />

## Benefits of Position Vectors

Position vectors have several benefits in mathematics and its applications:

1. Determining the location of a point in a coordinate system
2. Serving as a basis for calculating other vectors such as displacement vectors
3. Facilitating the solution of problems related to position and location
4. Used in GPS technology to determine the position of a location

## Relationship with Displacement Vectors

Displacement vectors can be obtained from the difference between two position vectors. If we have position vectors <InlineMath math="\overrightarrow{OA}" /> and <InlineMath math="\overrightarrow{OB}" />, then the displacement vector from A to B is:

<BlockMath math="\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}" />

<div className="mt-4">
  <Vector3d
    title="Relationship Between Position Vectors and Displacement Vector"
    description={
      <>
        Displacement vector <InlineMath math="AB" /> is obtained from the
        difference between position vectors <InlineMath math="OB" /> and{" "}
        <InlineMath math="OA" />.
      </>
    }
    vectors={[
      {
        from: [0, 0, 0],
        to: [-3, 2, 0],
        color: getColor("PINK"),
        label: "OA",
        labelPosition: "end",
      },
      {
        from: [0, 0, 0],
        to: [7, 5, 0],
        color: getColor("VIOLET"),
        label: "OB",
        labelPosition: "end",
      },
      {
        from: [-3, 2, 0],
        to: [7, 5, 0],
        color: getColor("TEAL"),
        label: "AB",
        labelPosition: "middle",
      },
    ]}
    cameraPosition={[10, 6, 10]}
  />
</div>

From the previous example, the displacement vector from A to B is:

<BlockMath
  math="\begin{align}
\overrightarrow{AB} &= \overrightarrow{OB} - \overrightarrow{OA} \\
&= (7, 5) - (-3, 2) \\
&= (7-(-3), 5-2) \\
&= (10, 3)
\end{align}"
/>

Therefore, to move from point A to point B, we need to move 10 units to the right and 3 units upward.
