The condition under which vector `(a+b)` and (a-b) should be at right angles to each other is :

Options

A. `a != b`
B. `a.b = 0`
C. `|a| = |b|`
D. `a.b = 1`

Answer

To determine the condition under which the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are at right angles to each other, we can use the concept of the dot product.

### Step-by-Step Solution:

1. **Understanding the Condition**:
For two vectors \( \mathbf{u} \) and \( \mathbf{v} \) to be at right angles, their dot product must be zero:
\[ \mathbf{u} \cdot \mathbf{v} = 0 \]

2. **Define the Vectors**:
Let \( \mathbf{u} = \mathbf{a} + \mathbf{b} \) and \( \mathbf{v} = \mathbf{a} - \mathbf{b} \).

3. **Calculate the Dot Product**:
We need to calculate:
\[ (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) \]

4. **Expand the Dot Product**:
Using the distributive property of the dot product, we expand:
\[ \mathbf{u} \cdot \mathbf{v} = \mathbf{a} \cdot \mathbf{a} - \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b} \]

5. **Simplify the Expression**:
Since \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \), the middle terms cancel out:
\[ \mathbf{u} \cdot \mathbf{v} = \mathbf{a} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b} \]
This simplifies to:
\[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{a}|^2 - |\mathbf{b}|^2 \]

6. **Set the Dot Product to Zero**:
For the vectors to be at right angles, we set the dot product to zero:
\[ |\mathbf{a}|^2 - |\mathbf{b}|^2 = 0 \]

7. **Solve for Magnitudes**:
This implies:
\[ |\mathbf{a}|^2 = |\mathbf{b}|^2 \]
Taking the square root gives:
\[ |\mathbf{a}| = |\mathbf{b}| \]

### Conclusion:
The condition under which the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are at right angles is that the magnitudes of the vectors \( \mathbf{a} \) and \( \mathbf{b} \) must be equal:
\[ |\mathbf{a}| = |\mathbf{b}| \]