2.12 Relative Position Vectors

Definition

A position vector describes the location of an object relative to the origin. Sometimes, however, we need to describe the position of one object relative to another object instead of the origin.

The vector that points from one object to another is called the relative position vector.

Unlike a position vector, a relative position vector does not start at the origin. It starts at the reference object and ends at the object whose position is being described.

Finding the Relative Position Vector

Suppose two objects, $\vec{A}$ and $\vec{B}$ , have position vectors $\vec{r}_A$ and $\vec{r}_B$ measured from the same origin.

The relative position vector from $\vec{A}$ to $\vec{B}$ is obtained by subtracting the position vector of $\vec{A}$ from the position vector of $\vec{B}$.

Relative position vector from A to B: $\vec{r}_{AB}=\vec{r}_B-\vec{r}_A$

Similarly,

Relative position vector from B to A: $\vec{r}_{BA}=\vec{r}_A-\vec{r}_B$

Notice that the two relative position vectors have the same magnitude but opposite directions.

Relative Position Using Coordinates

If the positions of two objects are given in coordinate form, the relative position vector is found by subtracting the corresponding coordinates.

Example

Object A is located at $\vec{A}=(2,1)$ and object B is located at $\vec{B}=(6,4)$.

The relative position vector from A to B is found by subtracting the coordinates:

x-component:

y-component:

Therefore, the relative position vector is

This means that object B is 4 m in the positive x-direction and 3 m in the positive y-direction relative to object A.

Position Vector vs Relative Position Vector

A position vector always begins at the origin and ends at the object’s position.

A relative position vector begins at one object and ends at another object. Although both are vectors, they describe different reference points.

Why Are Relative Position Vectors Important?

Relative position vectors allow us to describe the location of one object with respect to another object.

This idea is widely used in mechanics, navigation, robotics, astronomy, and engineering whenever the positions of multiple objects must be compared.

Key Points

• A relative position vector describes the position of one object relative to another object.
• It is obtained by subtracting one position vector from another.
• A relative position vector starts at one object and ends at another object.
• Reversing the order of the objects reverses the direction of the relative position vector.
• Relative position vectors can be calculated easily using coordinate subtraction.