In going from one city to another, a car travels 75`km` north, 60`km` north-west and 20`km` east,The magnitude of displacement between the two cities is `(take1//sqrt(2)=0.7)`

Options

A. 170`km`
B. 137`km`
C. 119`km`
D. 140`km`

Explanation

To find the magnitude of the displacement between the two cities based on the car's journey, we can break down the problem into steps:

### Step 1: Define the Directions
We will consider the eastward direction as the x-axis and the northward direction as the y-axis.

### Step 2: Represent the Movements as Vectors
1. **First Movement (75 km North)**:
- This can be represented as \( \vec{s_1} = 75 \hat{j} \) km.

2. **Second Movement (60 km North-West)**:
- North-West means the direction is at a 45-degree angle to both the north and west.
- The components can be calculated as:
- \( \text{North component} = 60 \cdot \frac{1}{\sqrt{2}} = 60 \cdot 0.7 = 42 \) km (in the north direction)
- \( \text{West component} = 60 \cdot \frac{1}{\sqrt{2}} = 60 \cdot 0.7 = 42 \) km (in the west direction)
- Therefore, this can be represented as \( \vec{s_2} = -42 \hat{i} + 42 \hat{j} \) km.

3. **Third Movement (20 km East)**:
- This can be represented as \( \vec{s_3} = 20 \hat{i} \) km.

### Step 3: Sum the Displacement Vectors
Now we can find the net displacement vector \( \vec{s_{net}} \) by adding the three vectors:
\[ \vec{s_{net}} = \vec{s_1} + \vec{s_2} + \vec{s_3} \]
Substituting the vectors:
\[ \vec{s_{net}} = (0 \hat{i} + 75 \hat{j}) + (-42 \hat{i} + 42 \hat{j}) + (20 \hat{i} + 0 \hat{j}) \]
Combining the components:
- **i-component**: \( 0 - 42 + 20 = -22 \)
- **j-component**: \( 75 + 42 + 0 = 117 \)

Thus, the net displacement vector is:
\[ \vec{s_{net}} = -22 \hat{i} + 117 \hat{j} \text{ km} \]

### Step 4: Calculate the Magnitude of the Displacement
The magnitude of the displacement can be calculated using the Pythagorean theorem:
\[ |\vec{s_{net}}| = \sqrt{(-22)^2 + (117)^2} \]
Calculating the squares:
\[ = \sqrt{484 + 13689} = \sqrt{14173} \]
Now, we can approximate \( \sqrt{14173} \):
Using the approximation \( 1/\sqrt{2} \approx 0.7 \):
\[ \sqrt{14173} \approx 119 \text{ km} \]

### Final Answer
The magnitude of the displacement between the two cities is approximately **119 km**.

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