The magnetitude of pairs of displacement vectors are given. Which pairs of displacement vector cannot be added to give a resultant vector of magnitude `13` cm?

Options

A. `4 cm, 16 cm`
B. `20 cm, 7 cm`
C. `1 cm, 15 cm`
D. `6 cm, 8 cm`

Explanation

To solve the problem of which pairs of displacement vectors cannot be added to give a resultant vector of magnitude 13 cm, we will follow these steps:

### Step-by-Step Solution

1. **Understanding the Range of Resultant Vector**:
The resultant vector \( R \) of two vectors \( A \) and \( B \) can be determined using the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
The maximum value of \( R \) occurs when \( \theta = 0^\circ \) (vectors are in the same direction), and the minimum value occurs when \( \theta = 180^\circ \) (vectors are in opposite directions).

2. **Calculating Maximum and Minimum Values**:
- **Maximum Value**:
\[ R_{\text{max}} = A + B \]
- **Minimum Value**:
\[ R_{\text{min}} = |A - B| \]

3. **Analyzing Each Pair of Vectors**:
We will analyze each pair of vectors given in the options to determine their maximum and minimum resultant magnitudes.

- **Pair 1: Magnitudes 4 cm and 6 cm**:
- Maximum: \( 4 + 6 = 10 \) cm
- Minimum: \( |4 - 6| = 2 \) cm
- Range: \( [2, 10] \) cm (13 cm is not in this range)

- **Pair 2: Magnitudes 7 cm and 8 cm**:
- Maximum: \( 7 + 8 = 15 \) cm
- Minimum: \( |7 - 8| = 1 \) cm
- Range: \( [1, 15] \) cm (13 cm is in this range)

- **Pair 3: Magnitudes 15 cm and 1 cm**:
- Maximum: \( 15 + 1 = 16 \) cm
- Minimum: \( |15 - 1| = 14 \) cm
- Range: \( [14, 16] \) cm (13 cm is not in this range)

- **Pair 4: Magnitudes 6 cm and 8 cm**:
- Maximum: \( 6 + 8 = 14 \) cm
- Minimum: \( |6 - 8| = 2 \) cm
- Range: \( [2, 14] \) cm (13 cm is in this range)

4. **Conclusion**:
The pairs of displacement vectors that cannot be added to give a resultant vector of magnitude 13 cm are:
- Pair 1: 4 cm and 6 cm
- Pair 3: 15 cm and 1 cm

### Final Answer:
The pairs of displacement vectors that cannot be added to give a resultant vector of magnitude 13 cm are:
- Pair 1: 4 cm and 6 cm
- Pair 3: 15 cm and 1 cm