O is apoint on the gournd chosen as origin. A boby first suffers a displacement of `10sqrt(2)`m North-East, next 10 m north and finally `10sqrt(2)` North-West. How far it is from the origin.

Options

A. 30n north
B. 30 m south
C. 30 m west
D. 30 m east

Explanation

To solve the problem step by step, we will break down the displacements into their vector components and then find the resultant displacement from the origin.

### Step 1: Understanding the Displacements
The body undergoes three displacements:
1. **10√2 m North-East**
2. **10 m North**
3. **10√2 m North-West**

### Step 2: Displacement in North-East Direction
The first displacement is 10√2 m in the North-East direction. In vector form, this can be broken down into its components:
- North-East means 45 degrees from the North and East axes.
- The components can be calculated using trigonometric functions:
- \( x \) (East) component: \( 10\sqrt{2} \cos(45^\circ) = 10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10 \) m
- \( y \) (North) component: \( 10\sqrt{2} \sin(45^\circ) = 10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10 \) m

Thus, the vector for this displacement is:
\[ \vec{OA} = 10 \hat{i} + 10 \hat{j} \]

### Step 3: Displacement in North Direction
The second displacement is 10 m North. In vector form, this is:
\[ \vec{AB} = 0 \hat{i} + 10 \hat{j} \]

### Step 4: Displacement in North-West Direction
The third displacement is 10√2 m in the North-West direction. This can also be broken down into components:
- North-West means 135 degrees from the East axis.
- The components can be calculated as:
- \( x \) (West) component: \( 10\sqrt{2} \cos(135^\circ) = 10\sqrt{2} \cdot \left(-\frac{1}{\sqrt{2}}\right) = -10 \) m
- \( y \) (North) component: \( 10\sqrt{2} \sin(135^\circ) = 10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10 \) m

Thus, the vector for this displacement is:
\[ \vec{BC} = -10 \hat{i} + 10 \hat{j} \]

### Step 5: Finding the Resultant Displacement
Now we can find the resultant displacement from the origin to point C by adding the vectors:
\[ \vec{OC} = \vec{OA} + \vec{AB} + \vec{BC} \]
Substituting the vectors:
\[ \vec{OC} = (10 \hat{i} + 10 \hat{j}) + (0 \hat{i} + 10 \hat{j}) + (-10 \hat{i} + 10 \hat{j}) \]
Combining like terms:
\[ \vec{OC} = (10 + 0 - 10) \hat{i} + (10 + 10 + 10) \hat{j} = 0 \hat{i} + 30 \hat{j} \]

### Step 6: Magnitude of the Resultant Displacement
The magnitude of the resultant displacement vector \( \vec{OC} \) is:
\[ |\vec{OC}| = \sqrt{(0)^2 + (30)^2} = 30 \text{ m} \]

### Step 7: Direction of the Resultant Displacement
The direction is along the positive y-axis, which corresponds to North.

### Final Answer
The body is **30 m North** from the origin.

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