KAHIBARO
Table of Contents
Locating an Object in Space
In motion in two and three dimensions, a single number is no longer enough to describe where an object is. We need a way to specify both how far the object is from the origin and in which direction it lies. The tool for this is the position vector.
A position vector is a vector that points from the chosen origin of a coordinate system to the location of the object. It tells us where the object is in space relative to that origin. If the object moves, its position vector changes with time.
Position Vector in Two Dimensions
In a two dimensional Cartesian plane, the position of a particle is described by its coordinates $(x, y)$. The corresponding position vector is written as
$$
\vec{r} = x \hat{i} + y \hat{j}
$$
Here, $\hat{i}$ is the unit vector in the positive $x$ direction, and $\hat{j}$ is the unit vector in the positive $y$ direction. The coefficients $x$ and $y$ are the components of the position vector.
This means that the particle is reached by moving $x$ units horizontally and $y$ units vertically from the origin.
Position Vector in Three Dimensions
In three dimensions, one more coordinate is needed. A point has coordinates $(x, y, z)$, and its position vector is
$$
\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}
$$
Here, $\hat{k}$ is the unit vector in the positive $z$ direction. The three components describe the location of the object in space.
For example, if a particle is at $(2, -1, 4)$, then its position vector is
$$
\vec{r} = 2\hat{i} - \hat{j} + 4\hat{k}
$$
This gives a complete description of where the particle is relative to the origin.
Magnitude of the Position Vector
The magnitude of the position vector gives the straight line distance from the origin to the object. In two dimensions,
$$
|\vec{r}| = \sqrt{x^2 + y^2}
$$
and in three dimensions,
$$
|\vec{r}| = \sqrt{x^2 + y^2 + z^2}
$$
This is just the Pythagorean theorem extended to vectors.
Important formulas for position vectors:
$$
\vec{r} = x\hat{i} + y\hat{j}
$$
$$
\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}
$$
$$
|\vec{r}| = \sqrt{x^2 + y^2}
$$
$$
|\vec{r}| = \sqrt{x^2 + y^2 + z^2}
$$
A position vector always starts at the origin and ends at the point representing the object's location.
Position Vector as a Function of Time
When an object moves, its coordinates change with time. Then the position vector also depends on time:
$$
\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}
$$
in two dimensions, or
$$
\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}
$$
in three dimensions.
This notation emphasizes that the position is not fixed. At each instant $t$, the object has a particular location in space.
For example, if
$$
x(t) = 2t, \qquad y(t) = t^2
$$
then the position vector is
$$
\vec{r}(t) = 2t\,\hat{i} + t^2\,\hat{j}
$$
At $t = 1$,
$$
\vec{r}(1) = 2\hat{i} + \hat{j}
$$
and at $t = 2$,
$$
\vec{r}(2) = 4\hat{i} + 4\hat{j}
$$
So the vector changes as the particle moves.
Interpreting the Components
Each component of the position vector measures location along one coordinate axis. A positive value means the object lies in the positive direction of that axis, and a negative value means it lies in the negative direction.
The table below summarizes this idea.
| Component | Meaning |
|---|---|
| $x$ | Horizontal position along the $x$ axis |
| $y$ | Vertical or sideways position along the $y$ axis |
| $z$ | Position along the $z$ axis in three dimensional space |
The full position vector combines all of these into one quantity that includes both distance and direction.
Position Vector and Path
The position vector tells us where the object is at a particular instant. As time changes, the tip of the vector traces out the path of the object. So the path or trajectory can be viewed as the set of all positions $\vec{r}(t)$.
This is useful because motion in two and three dimensions is naturally described by following how the position vector changes over time.
Choosing the Origin
A position vector always depends on the choice of origin. If the origin changes, the position vector changes too, even though the object remains at the same physical place.
This means position is relative. There is no single absolute position in mechanics. What matters is the location of the object with respect to the coordinate system we choose.
A position vector does not describe an object's size or shape. It only describes the location of one point, usually the particle or the center of an object.
Simple Example
Suppose a drone is located at the point $(3, 4, 2)$. Its position vector is
$$
\vec{r} = 3\hat{i} + 4\hat{j} + 2\hat{k}
$$
Its distance from the origin is
$$
|\vec{r}| = \sqrt{3^2 + 4^2 + 2^2} = \sqrt{29}
$$
So the vector tells us both where the drone is and how far it is from the origin.
Main Idea
The position vector is the basic way to represent location in two and three dimensional motion. It points from the origin to the object, its components give the coordinates, and its magnitude gives the distance from the origin. Once position is written as $\vec{r}(t)$, the motion of an object can be described mathematically in a clear and compact form.