A body A is standing `20sqrt(3)` m away in a direction `30^(@)` north of east from east. In a Cartesian coordinate system with its x-axis towards the east, the position of C with respect to A is

Options

A. `(-20hati+ -10hatj)m`
B. `(-10hati-10sqrt(3)hatj)m`
C. `(10hati+10sqrt(3)hatj)m`
D. It depends on where we chose the origin.

Answer

To solve the problem, we need to determine the position of body C with respect to body A, given that body A is located at a distance of \(20\sqrt{3}\) m in a direction of \(30^\circ\) north of east. We will use the Cartesian coordinate system where the x-axis points east and the y-axis points north.

### Step-by-Step Solution:

1. **Understanding the Position of A**:
- The position of body A can be represented in a Cartesian coordinate system. The distance from the origin (which we can assume is point O) to A is \(20\sqrt{3}\) m at an angle of \(30^\circ\) north of east.

2. **Calculating the Coordinates of A**:
- The x-coordinate (east direction) can be calculated using the cosine of the angle:
\[ x_A = r \cdot \cos(\theta) = 20\sqrt{3} \cdot \cos(30^\circ) \]
- The y-coordinate (north direction) can be calculated using the sine of the angle:
\[ y_A = r \cdot \sin(\theta) = 20\sqrt{3} \cdot \sin(30^\circ) \]

3. **Substituting the Values**:
- We know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) and \(\sin(30^\circ) = \frac{1}{2}\).
- Therefore:
\[ x_A = 20\sqrt{3} \cdot \frac{\sqrt{3}}{2} = 20 \cdot \frac{3}{2} = 30 \text{ m} \]
\[ y_A = 20\sqrt{3} \cdot \frac{1}{2} = 10\sqrt{3} \text{ m} \]

4. **Position of A in Cartesian Coordinates**:
- The coordinates of A are:
\[ A(30, 10\sqrt{3}) \]

5. **Position of C with Respect to A**:
- Since the problem does not provide any specific coordinates for C, we can state that the position of C with respect to A depends on the origin we choose. If we consider A as the new origin, then the position of C relative to A can be expressed as:
\[ C = (x_C - x_A, y_C - y_A) \]
- However, without specific coordinates for C, we cannot determine its exact position.

### Conclusion:
The position of C with respect to A is dependent on the coordinates of C, which are not provided in the question. Therefore, the position of C can only be defined relative to A based on the chosen origin.