A man first moves 3 m due east , them 6 m due north and finally 7 m due west , then the magnitude of the resultant displacement is (in metre)

Options

A. `sqrt(16)`
B. `sqrt(24)`
C. `sqrt(52)`
D. `sqrt(94)`

Explanation

To solve the problem step by step, we will analyze the man's movements and calculate the resultant displacement.

### Step 1: Define the movements
The man moves in three different directions:
1. **3 meters due east**: This can be represented as \( +3 \hat{i} \).
2. **6 meters due north**: This can be represented as \( +6 \hat{j} \).
3. **7 meters due west**: This can be represented as \( -7 \hat{i} \).

### Step 2: Write the displacement vectors
Now we can express the total displacement vector by combining these movements:
- East and West movements are along the x-axis (i direction).
- North movement is along the y-axis (j direction).

The total displacement vector \( \vec{D} \) can be calculated as:
\[ \vec{D} = (3 \hat{i} + 6 \hat{j} - 7 \hat{i}) \]

### Step 3: Combine the i components
Now, we combine the i components:
\[ \vec{D} = (3 - 7) \hat{i} + 6 \hat{j} = -4 \hat{i} + 6 \hat{j} \]

### Step 4: Calculate the magnitude of the resultant displacement
The magnitude of the resultant displacement can be calculated using the Pythagorean theorem:
\[ |\vec{D}| = \sqrt{(-4)^2 + (6)^2} \]
Calculating the squares:
\[ |\vec{D}| = \sqrt{16 + 36} = \sqrt{52} \]

### Step 5: Simplify the result
We can simplify \( \sqrt{52} \):
\[ \sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13} \]

### Final Answer
Thus, the magnitude of the resultant displacement is \( 2\sqrt{13} \) meters.

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