What is the magnitude of the resultant vector when adding (a+b+c), when a is 25 N at 0 degrees, b is 35 N at 180 degrees, and c is 20 N at 270 degrees?

To find the magnitude of the resultant vector when adding vectors \(a + b + c\), we need to resolve each vector into its components and then add them accordingly.

### Step 1: Resolve each vector into components
- **Vector \(a\)**: Magnitude = 25 N, Angle = \(0^\circ\)
    - \(a_x = 25 \cos(0^\circ) = 25\) N
    - \(a_y = 25 \sin(0^\circ) = 0\) N

- **Vector \(b\)**: Magnitude = 35 N, Angle = \(180^\circ\)
    - \(b_x = 35 \cos(180^\circ) = -35\) N
    - \(b_y = 35 \sin(180^\circ) = 0\) N

- **Vector \(c\)**: Magnitude = 20 N, Angle = \(270^\circ\)
    - \(c_x = 20 \cos(270^\circ) = 0\) N
    - \(c_y = 20 \sin(270^\circ) = -20\) N

### Step 2: Sum the components
Now we add the corresponding components for the total vector:

- \( R_x = a_x + b_x + c_x = 25 + (-35) + 0 = -10\) N
- \( R_y = a_y + b_y + c_y = 0 + 0 + (-20) = -20\) N

### Step 3: Calculate the magnitude of the resultant vector
The magnitude of the resultant vector \(R\) is given by:
\[
|R| = \sqrt{R_x^2 + R_y^2}
\]
Substituting the values:
\[
|R| = \sqrt{(-10)^2 + (-20)^2} = \sqrt{100 + 400} = \sqrt{500} \approx 22.36 \, \text{N}
\]

### Conclusion:
The magnitude of the resultant vector is approximately **22.36 N**.

Would you like more details on this calculation or any further questions?

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Here are 5 related questions:
1. How do you determine the direction of the resultant vector?
2. What are vector components, and why are they important?
3. How do you calculate the angle of a resultant vector?
4. What happens if two vectors have the same magnitude but opposite directions?
5. How do you add vectors graphically?

**Tip:** Always resolve vectors into components to simplify addition!

Learn how to calculate the magnitude of the resultant vector when adding vectors a, b, and c at different angles (25 N at 0°, 35 N at 180°, and 20 N at 270°). This solution involves vector components and the Pythagorean theorem, making it suitable for high school students in Grades 9-12.

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