What is the magnitude and direction of the resultant vector when adding (a+b+c), when a is 20 N at 0 degrees, b is 25 N at 180 degrees, and c is 5 N at 180 degrees?

To find the magnitude and direction of the resultant vector when adding \( a + b + c \), we'll break each vector into its horizontal (x-axis) and vertical (y-axis) components. Since all vectors are along the horizontal axis, the vertical components will be zero.

### Given vectors:
- \( \vec{a} = 20 \, \text{N at} \, 0^\circ \)
- \( \vec{b} = 25 \, \text{N at} \, 180^\circ \)
- \( \vec{c} = 5 \, \text{N at} \, 180^\circ \)

### Step 1: Break the vectors into components.

1. **For \( \vec{a} \):**
   - \( a_x = 20 \, \text{N} \) (since it's along the positive x-axis, \( 0^\circ \))
   - \( a_y = 0 \, \text{N} \) (no vertical component)

2. **For \( \vec{b} \):**
   - \( b_x = -25 \, \text{N} \) (since it's along the negative x-axis, \( 180^\circ \))
   - \( b_y = 0 \, \text{N} \)

3. **For \( \vec{c} \):**
   - \( c_x = -5 \, \text{N} \) (also along the negative x-axis, \( 180^\circ \))
   - \( c_y = 0 \, \text{N} \)

### Step 2: Add the components.

- \( R_x = a_x + b_x + c_x = 20 + (-25) + (-5) = -10 \, \text{N} \)
- \( R_y = 0 + 0 + 0 = 0 \, \text{N} \)

### Step 3: Calculate the magnitude of the resultant vector.

Since there is no vertical component, the magnitude \( |R| \) is simply the absolute value of the horizontal component:

\[
|R| = |R_x| = 10 \, \text{N}
\]

### Step 4: Determine the direction.

Since the resultant is entirely in the negative x-direction, the angle is \( 180^\circ \).

### Final Answer:
- **Magnitude**: \( 10 \, \text{N} \)
- **Direction**: \( 180^\circ \)

Would you like any further details or clarifications?

Here are five related questions to explore:
1. What is the resultant when vectors are added at different angles?
2. How do you find the resultant of three vectors in 2D space?
3. How do trigonometric functions help resolve vectors into components?
4. How does vector addition change in 3D compared to 2D?
5. Can you explain how to add two vectors graphically?

**Tip**: Always break vectors into components when adding vectors at different angles to simplify calculations.

Discover how to calculate the magnitude and direction of the resultant vector when adding vectors with magnitudes of 20 N, 25 N, and 5 N at different angles. This step-by-step solution demonstrates vector component addition and is suitable for high school students.

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