If `overset(rarr)A` and `overset(rarr)B` are two vectors of non-zero magnitude, which of the following relations cannot be possible ?

Options

A. `overset(rarr)A +overset(rarr)B=overset(rarr)A-overset(rarr)B`
B. `overset(rarr)A.overset(rarr)B=|overset(rarr)A xxoverset(rarr)B|`
C. `|overset(rarr)A+overset(rarr)B|=|overset(rarr)A-overset(rarr)B|`
D. `|overset(rarr)A+overset(rarr)B| gt |overset(rarr)A|+|overset(rarr)B|`

Explanation

To determine which of the given relations involving the vectors \( \overset{\rarr}{A} \) and \( \overset{\rarr}{B} \) cannot be possible, we will analyze each relation step by step.

### Step 1: Understanding the Vectors
Let \( \overset{\rarr}{A} \) and \( \overset{\rarr}{B} \) be two vectors with non-zero magnitudes. We denote their magnitudes as \( |\overset{\rarr}{A}| = a \) and \( |\overset{\rarr}{B}| = b \), where \( a, b > 0 \).

### Step 2: Analyzing the Relations
We need to analyze the following relations:

1. \( \overset{\rarr}{A} + \overset{\rarr}{B} = |\overset{\rarr}{A}| - |\overset{\rarr}{B}| \)
2. \( |\overset{\rarr}{A}| + |\overset{\rarr}{B}| = |\overset{\rarr}{A} - \overset{\rarr}{B}| \)
3. \( \overset{\rarr}{A} \cdot \overset{\rarr}{B} = |\overset{\rarr}{A}| |\overset{\rarr}{B}| \cos(\theta) \)
4. \( |\overset{\rarr}{A} + \overset{\rarr}{B}| < |\overset{\rarr}{A}| + |\overset{\rarr}{B}| \)

### Step 3: Evaluating Each Relation
1. **Relation 1: \( \overset{\rarr}{A} + \overset{\rarr}{B} = |\overset{\rarr}{A}| - |\overset{\rarr}{B}| \)**
- This relation implies that the resultant vector \( \overset{\rarr}{A} + \overset{\rarr}{B} \) has a magnitude equal to the difference of the magnitudes of \( \overset{\rarr}{A} \) and \( \overset{\rarr}{B} \). This is not possible because the magnitude of a vector cannot be negative, and the sum of two vectors cannot be less than the magnitude of either vector. Therefore, this relation **cannot be possible**.

2. **Relation 2: \( |\overset{\rarr}{A}| + |\overset{\rarr}{B}| = |\overset{\rarr}{A} - \overset{\rarr}{B}| \)**
- This relation is valid only when the two vectors are in opposite directions and have equal magnitudes. Thus, it can be possible under specific conditions.

3. **Relation 3: \( \overset{\rarr}{A} \cdot \overset{\rarr}{B} = |\overset{\rarr}{A}| |\overset{\rarr}{B}| \cos(\theta) \)**
- This is the definition of the dot product of two vectors and is always valid for any angle \( \theta \).

4. **Relation 4: \( |\overset{\rarr}{A} + \overset{\rarr}{B}| < |\overset{\rarr}{A}| + |\overset{\rarr}{B}| \)**
- This relation is a statement of the triangle inequality, which is always true for any two vectors.

### Conclusion
Based on the analysis above, the relations that cannot be possible are:
- **Relation 1: \( \overset{\rarr}{A} + \overset{\rarr}{B} = |\overset{\rarr}{A}| - |\overset{\rarr}{B}| \)**