On a horizontal flat ground, a person is standing at a point `A`. At this point, he installs a `5m` long pole vertically. Now , he moves `5m` towards east and then `2m` towards north and reaches at a point `B`. There he installs another `3m` long vertical pole . A bird flies from the top of the first pole to the top of the swecond pole. Find the displacement and magnitude of the displacement of the bird.

Explanation

To solve the problem step by step, we will break down the movements and calculate the displacement of the bird flying from the top of the first pole to the top of the second pole.

### Step 1: Define the Points and Poles
1. **Point A**: The position where the first pole is installed.
- Height of the pole at A = 5 m (vertical)
- Coordinates of point A = (0, 0, 5) (x, y, z)

2. **Movement to Point B**:
- The person moves 5 m towards the east (positive x-direction).
- Then, he moves 2 m towards the north (positive y-direction).
- Therefore, the coordinates of point B = (5, 2, 3) (since the height of the second pole is 3 m).

### Step 2: Calculate the Displacement Vector
The displacement vector from the top of the first pole (Point P) to the top of the second pole (Point Q) can be calculated as follows:

- **Coordinates of Point P (top of the first pole)**: (0, 0, 5)
- **Coordinates of Point Q (top of the second pole)**: (5, 2, 3)

The displacement vector \( \vec{PQ} \) can be calculated using the formula:
\[ \vec{PQ} = \vec{Q} - \vec{P} \]
Substituting the coordinates:
\[ \vec{PQ} = (5, 2, 3) - (0, 0, 5) \]
Breaking it down into components:
\[ \vec{PQ} = (5 - 0, 2 - 0, 3 - 5) = (5, 2, -2) \]

### Step 3: Calculate the Magnitude of the Displacement
The magnitude of the displacement vector \( \vec{PQ} \) can be calculated using the formula:
\[ |\vec{PQ}| = \sqrt{(x^2 + y^2 + z^2)} \]
Substituting the components:
\[ |\vec{PQ}| = \sqrt{(5^2 + 2^2 + (-2)^2)} = \sqrt{(25 + 4 + 4)} = \sqrt{33} \]

### Final Answers
- **Displacement Vector**: \( \vec{PQ} = 5\hat{i} + 2\hat{j} - 2\hat{k} \)
- **Magnitude of Displacement**: \( |\vec{PQ}| = \sqrt{33} \, \text{m} \)