Answer:
The position vector \(\mathbf{r}\) of a particle located in a plane with reference to the origin of an \(xy\) reference frame is given by:
\[
\mathbf{r} = X \hat{i} + Y \hat{j}
\]
Here's a breakdown of what \(X\) and \(Y\) represent:
1. **\(X\)**: This is the x-coordinate of the particle's position in the plane. It specifies the particle's horizontal displacement from the origin along the x-axis.
2. **\(Y\)**: This is the y-coordinate of the particle's position in the plane. It specifies the particle's vertical displacement from the origin along the y-axis.
### Vector Representation Diagrammatically
To represent this vector diagrammatically:
1. Draw an \(xy\) coordinate plane.
2. Plot the point \((X, Y)\) on this plane.
3. Draw a vector \(\mathbf{r}\) starting from the origin (0,0) to the point \((X, Y)\).
Here's a simple diagram to illustrate this:
```plaintext
Y-axis
|
| * (X, Y)
| /
| /
| /
| /
| /
| /
| /
| /
| /
|/
---------------------------- X-axis
(0,0)
```
In this diagram:
- The origin \((0,0)\) is the starting point of the vector.
- The point \((X, Y)\) is the endpoint of the vector.
- The vector \(\mathbf{r}\) is shown as the arrow from the origin to the point \((X, Y)\).
This visually demonstrates the position of the particle in the \(xy\) plane.