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URL: https://nakafa.com/en/subjects/mathematics/vector-operations/position-vector
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/vector-operations/position-vector/en.mdx

Learn position vectors from origin to points. Learn coordinate representation, displacement relationships, GPS applications, and coordinate-space visualization.

---

## 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.

Visible text: A position vector is a vector that starts from point (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.

Component: Vector3d
Props:
- title: Position Vector Visualization
- description: Examples of position vectors from origin $$O$$ to points{" "}
$$A$$ and $$B$$.
  Visible text: Examples of position vectors from origin to points{" "}
 and .
- 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

Visible text: - Always starts from the origin (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 $$(x, y)$$ in a plane, then the position vector from point $$O$$ to point $$P$$ can be written as $$\overrightarrow{OP} = (x, y)$$.

Visible text: In general, if we have a point with coordinates in a plane, then the position vector from point to point can be written as .

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

Visible text: In three-dimensional space, if point has coordinates , then its position vector is .

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 $$D$$).

Visible text: In the visualization below, we use the notation , , , and to indicate position vectors from point to specific points (, , , or ).

Component: Vector3d
Props:
- title: Position Vectors in Three-Dimensional 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:

Visible text: Suppose there are two points and in the coordinate plane:

- Point $$A$$ with coordinates $$(-3, 2)$$
- Point $$B$$ with coordinates $$(7, 5)$$

Visible text: - Point with coordinates 
- Point with coordinates

Then the position vectors of these two points are:

- $$\overrightarrow{OA} = (-3, 2)$$
- $$\overrightarrow{OB} = (7, 5)$$

Visible text: - 
-

## 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 $$\overrightarrow{OA}$$ and $$\overrightarrow{OB}$$, then the displacement vector from $$A$$ to $$B$$ is:

Visible text: Displacement vectors can be obtained from the difference between two position vectors. If we have position vectors and , then the displacement vector from to is:

```math
\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}
```

Component: ContentBlock
Children:
Component: Vector3d
Props:
- title: Relationship Between Position Vectors and Displacement Vector
- description: Displacement vector $$AB$$ is obtained from the
difference between position vectors $$OB$$ and{" "}
$$OA$$.
  Visible text: Displacement vector is obtained from the
difference between position vectors and{" "}
.
- 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]

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

Visible text: From the previous example, the displacement vector from to is:

```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 $$10 \text{ units}$$ to the right and $$3 \text{ units}$$ upward.

Visible text: Therefore, to move from point to point , we need to move to the right and upward.