Position Vector

Locating a Point from the Origin

A position vector starts at the origin $O$ and ends at a point in the coordinate system. Its components are the coordinates of that endpoint, so it records the point's location relative to the origin.

Characteristics of Position Vectors

All three properties come from the same fact: the tail is fixed at the origin $O$, so the endpoint alone determines the vector.

• 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:

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

Uses of Position Vectors

The uses fall into two groups: reading a single location, and comparing two locations.

1. Determine the location of a point in a coordinate system
2. Calculate displacement from the difference between two positions
3. Express changes of position with vector operations
4. Represent a location relative to a chosen coordinate origin, including coordinates derived from navigation systems such as GPS

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:

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