Consider the following four force vectors:
`vecF_(1) = 50.0N,` due east, `vecF_(2)= 10.0N,` due east,
`vecF _(3) =40.0N,` due west,` vecF_(4)=30.0N,` due west
Which two vectors add together to give a resultant with smallest magnitude ? In the option below, the two vectors are given followed by a magnitude and direction.

Options

A. `vecF _(1) + vecF_(3), 10- N,` due east
B. `vecF_(1) + vecF_(3), 10 N,` due east
C. `vecF_(1) + vecF_(2), 20 N` due east
D. `vecF_(1) + vecF_(4),20 N` due west

Answer

To solve the problem of which two force vectors add together to give a resultant with the smallest magnitude, we will follow these steps:

### Step 1: Identify the Force Vectors
We have the following force vectors:
- \( \vec{F_1} = 50.0 \, \text{N} \) (due east)
- \( \vec{F_2} = 10.0 \, \text{N} \) (due east)
- \( \vec{F_3} = 40.0 \, \text{N} \) (due west)
- \( \vec{F_4} = 30.0 \, \text{N} \) (due west)

### Step 2: Assign Directions
We can represent east as positive and west as negative. Thus:
- \( \vec{F_1} = +50.0 \, \text{N} \)
- \( \vec{F_2} = +10.0 \, \text{N} \)
- \( \vec{F_3} = -40.0 \, \text{N} \)
- \( \vec{F_4} = -30.0 \, \text{N} \)

### Step 3: Calculate Resultants for Each Pair
We will calculate the resultant for each possible pair of vectors:

1. **Pair \( \vec{F_1} \) and \( \vec{F_2} \)**:
\[ R_{12} = \vec{F_1} + \vec{F_2} = 50.0 + 10.0 = 60.0 \, \text{N} \quad (\text{east}) \]

2. **Pair \( \vec{F_1} \) and \( \vec{F_3} \)**:
\[ R_{13} = \vec{F_1} + \vec{F_3} = 50.0 - 40.0 = 10.0 \, \text{N} \quad (\text{east}) \]

3. **Pair \( \vec{F_1} \) and \( \vec{F_4} \)**:
\[ R_{14} = \vec{F_1} + \vec{F_4} = 50.0 - 30.0 = 20.0 \, \text{N} \quad (\text{east}) \]

4. **Pair \( \vec{F_2} \) and \( \vec{F_3} \)**:
\[ R_{23} = \vec{F_2} + \vec{F_3} = 10.0 - 40.0 = -30.0 \, \text{N} \quad (\text{west}) \]

5. **Pair \( \vec{F_2} \) and \( \vec{F_4} \)**:
\[ R_{24} = \vec{F_2} + \vec{F_4} = 10.0 - 30.0 = -20.0 \, \text{N} \quad (\text{west}) \]

6. **Pair \( \vec{F_3} \) and \( \vec{F_4} \)**:
\[ R_{34} = \vec{F_3} + \vec{F_4} = -40.0 - 30.0 = -70.0 \, \text{N} \quad (\text{west}) \]

### Step 4: Compare Resultant Magnitudes
Now we compare the magnitudes of the resultant vectors:
- \( |R_{12}| = 60.0 \, \text{N} \)
- \( |R_{13}| = 10.0 \, \text{N} \)
- \( |R_{14}| = 20.0 \, \text{N} \)
- \( |R_{23}| = 30.0 \, \text{N} \)
- \( |R_{24}| = 20.0 \, \text{N} \)
- \( |R_{34}| = 70.0 \, \text{N} \)

The smallest magnitude is \( 10.0 \, \text{N} \) from the pair \( \vec{F_1} \) and \( \vec{F_3} \).

### Conclusion
The two vectors that add together to give a resultant with the smallest magnitude are:
- \( \vec{F_1} \) and \( \vec{F_3} \)
- Resultant magnitude: \( 10.0 \, \text{N} \) (due east)

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