The condition under which the vector (a+b) and (a-b) are parallel is

Options

A. `abot b`
B. `|a|=|b|`
C. `a ne b`
D. `A || B`

Explanation

To determine the condition under which the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are parallel, we need to analyze the relationship between these two vectors.

### Step-by-Step Solution:

1. **Understanding Parallel Vectors**:
- Two vectors are parallel if the angle between them is either \( 0^\circ \) or \( 180^\circ \). This can be mathematically expressed using the cross product: two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are parallel if \( \mathbf{u} \times \mathbf{v} = \mathbf{0} \).

2. **Setting Up the Problem**:
- We need to check when \( (\mathbf{a} + \mathbf{b}) \) and \( (\mathbf{a} - \mathbf{b}) \) are parallel. Therefore, we calculate the cross product:
\[ (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b}) = \mathbf{0} \]

3. **Expanding the Cross Product**:
- Using the distributive property of the cross product:
\[ (\mathbf{a} + \mathbf{b}) \times (\mathbf{a} - \mathbf{b}) = \mathbf{a} \times \mathbf{a} - \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} - \mathbf{b} \times \mathbf{b} \]
- Since the cross product of any vector with itself is zero, we have:
\[ \mathbf{a} \times \mathbf{a} = \mathbf{0} \quad \text{and} \quad \mathbf{b} \times \mathbf{b} = \mathbf{0} \]
- Thus, the expression simplifies to:
\[ -\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} = -\mathbf{a} \times \mathbf{b} - \mathbf{a} \times \mathbf{b} = -2(\mathbf{a} \times \mathbf{b}) \]

4. **Setting the Cross Product to Zero**:
- For the vectors to be parallel, we need:
\[ -2(\mathbf{a} \times \mathbf{b}) = \mathbf{0} \]
- This implies:
\[ \mathbf{a} \times \mathbf{b} = \mathbf{0} \]

5. **Conclusion**:
- The condition \( \mathbf{a} \times \mathbf{b} = \mathbf{0} \) means that vectors \( \mathbf{a} \) and \( \mathbf{b} \) must be parallel. Therefore, the correct condition under which \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are parallel is:
\[ \mathbf{a} \parallel \mathbf{b} \]

### Final Answer:
The vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are parallel if \( \mathbf{a} \) is parallel to \( \mathbf{b} \).