A person moves `30 m` north, then `30 m`, then `20 m` towards east and finally `30sqrt(2) m` in south-west direction. The displacement of the person from the origin will be

Options

A. `10 m` along north
B. `10 m` long south
C. `10 m`along west
D. Zero

Explanation

To solve the problem step by step, we will break down the movements of the person and calculate the resultant displacement vector from the origin.

### Step 1: Define the movements
1. The person moves **30 m north**.
2. Then, they move **30 m east**.
3. Next, they move **20 m east**.
4. Finally, they move **30√2 m in the southwest direction**.

### Step 2: Represent movements as vectors
We will represent each movement as a vector in a Cartesian coordinate system where:
- North is represented by the positive y-axis (J-cap).
- East is represented by the positive x-axis (I-cap).
- South-west is at a 45-degree angle towards the south-west quadrant.

1. **Movement 1 (North)**:
\[ \text{Vector OA} = 30 \hat{j} \]

2. **Movement 2 (East)**:
\[ \text{Vector AB} = 30 \hat{i} \]

3. **Movement 3 (East)**:
\[ \text{Vector BC} = 20 \hat{i} \]

4. **Movement 4 (South-West)**:
The south-west direction can be broken down into components. Since it is at a 45-degree angle:
\[ \text{Magnitude} = 30\sqrt{2} \]
The components will be:
- In the x-direction (west): \(-30\) (since west is negative x)
- In the y-direction (south): \(-30\) (since south is negative y)

Therefore, we can write:
\[ \text{Vector CD} = -30 \hat{i} - 30 \hat{j} \]

### Step 3: Calculate the resultant vector
Now, we will sum all the vectors to find the resultant displacement vector from the origin to the final position.

1. **Total displacement vector**:
\[ \text{OC} = \text{OA} + \text{AB} + \text{BC} + \text{CD} \]
Substituting the vectors:
\[ \text{OC} = (30 \hat{j}) + (30 \hat{i}) + (20 \hat{i}) + (-30 \hat{i} - 30 \hat{j}) \]

Combining the components:
- In the x-direction:
\[ 30 + 20 - 30 = 20 \hat{i} \]
- In the y-direction:
\[ 30 - 30 = 0 \hat{j} \]

Thus, the resultant vector is:
\[ \text{OC} = 20 \hat{i} \]

### Step 4: Calculate the magnitude of the displacement
The magnitude of the displacement vector is simply the length of the vector:
\[ |\text{OC}| = \sqrt{(20)^2 + (0)^2} = 20 \text{ m} \]

### Step 5: Direction of the displacement
Since the resultant vector has no y-component, the direction is purely along the x-axis, which is towards the east.

### Final Answer:
The displacement of the person from the origin is **20 m towards the east**.

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