A man walks 30 m towards north, then 20 m, towards east and in the last `30sqrt(2)` m towards south - west. The displacement from origin is :

Options

A. 10m towards west
B. 10 m towards east
C. `60sqrt(2)` m towards north west
D. `60sqrt(2)` m towards east north

Answer

To solve the problem, we need to determine the displacement of a man who walks in three different directions. We will break down the movements into vectors and then calculate the resultant displacement.

### Step-by-Step Solution:

1. **Identify the Movements**:
- The man walks 30 m North.
- Then he walks 20 m East.
- Finally, he walks \(30\sqrt{2}\) m towards South-West.

2. **Convert Movements into Vectors**:
- The North direction can be represented as the positive y-axis, and East as the positive x-axis.
- The South-West direction is at a 45-degree angle to both the South and West axes.

3. **Define the Vectors**:
- **A vector (North)**:
\[ \vec{A} = 30 \hat{j} \text{ m} \]
- **B vector (East)**:
\[ \vec{B} = 20 \hat{i} \text{ m} \]
- **C vector (South-West)**:
- The components of the South-West movement can be calculated as follows:
\[ \text{Magnitude of C} = 30\sqrt{2} \text{ m} \]
- Since South-West is at a 45-degree angle, both the x and y components will be equal:
\[ C_x = C_y = \frac{30\sqrt{2}}{\sqrt{2}} = 30 \text{ m} \]
- The x-component (West) will be negative and the y-component (South) will also be negative:
\[ \vec{C} = -30 \hat{i} - 30 \hat{j} \text{ m} \]

4. **Combine the Vectors**:
- Now, we will sum the vectors to find the resultant displacement vector:
\[ \vec{R} = \vec{A} + \vec{B} + \vec{C} \]
- Substituting the vectors:
\[ \vec{R} = (30 \hat{j}) + (20 \hat{i}) + (-30 \hat{i} - 30 \hat{j}) \]
- Simplifying:
\[ \vec{R} = (20 - 30) \hat{i} + (30 - 30) \hat{j} = -10 \hat{i} + 0 \hat{j} \]
- Thus, the resultant vector is:
\[ \vec{R} = -10 \hat{i} \text{ m} \]

5. **Magnitude and Direction**:
- The magnitude of the displacement is:
\[ |\vec{R}| = 10 \text{ m} \]
- The direction is towards the West since the x-component is negative.

### Final Answer:
The displacement from the origin is **10 m towards the West**.

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