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 determine which pairs of displacement vectors cannot be added to give a resultant vector of magnitude 13 cm, we will use the triangle inequality theorem. According to this theorem, for two vectors A and B, the magnitude of their resultant R must satisfy the following conditions:

1. \( |A - B| \leq R \leq |A + B| \)

Where:
- \( |A - B| \) is the minimum possible resultant.
- \( |A + B| \) is the maximum possible resultant.

We will analyze each pair of displacement vectors provided in the options.

### Step 1: Analyze Each Option

**Option A: Magnitudes 4 cm and 12 cm**
- Minimum resultant: \( |12 - 4| = 8 \) cm
- Maximum resultant: \( |12 + 4| = 16 \) cm
- Condition: \( 8 \leq R \leq 16 \)
- Since 13 cm lies within this range, this pair can give a resultant of 13 cm.

**Option B: Magnitudes 7 cm and 20 cm**
- Minimum resultant: \( |20 - 7| = 13 \) cm
- Maximum resultant: \( |20 + 7| = 27 \) cm
- Condition: \( 13 \leq R \leq 27 \)
- Since 13 cm equals the minimum resultant, this pair can also give a resultant of 13 cm.

**Option C: Magnitudes 1 cm and 15 cm**
- Minimum resultant: \( |15 - 1| = 14 \) cm
- Maximum resultant: \( |15 + 1| = 16 \) cm
- Condition: \( 14 \leq R \leq 16 \)
- Since 13 cm is less than the minimum resultant (14 cm), this pair cannot give a resultant of 13 cm.

**Option D: Magnitudes 6 cm and 8 cm**
- Minimum resultant: \( |8 - 6| = 2 \) cm
- Maximum resultant: \( |8 + 6| = 14 \) cm
- Condition: \( 2 \leq R \leq 14 \)
- Since 13 cm lies within this range, this pair can give a resultant of 13 cm.

### Conclusion
The pair of displacement vectors that cannot be added to give a resultant vector of magnitude 13 cm is **Option C: 1 cm and 15 cm**.