Position Vector

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.

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

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

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 $D$).

Examples of Position Vectors

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

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

Then the position vectors of these two points are:

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

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

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

Therefore, to move from point $A$ to point $B$, we need to move $10 \text{ units}$ to the right and $3 \text{ units}$ upward.