If `|a| = |b|`, then `(a +b). (a-b)` is

Options

A. `+` ive
B. `-`ive
C. zero
D. none of these

Answer

To solve the problem, we need to evaluate the expression \((a + b) \cdot (a - b)\) given that \(|a| = |b|\).

### Step-by-Step Solution:

1. **Write the expression**:
We start with the expression:
\[ (a + b) \cdot (a - b) \]

2. **Apply the distributive property of the dot product**:
Using the distributive property, we can expand the expression:
\[ (a + b) \cdot (a - b) = a \cdot a - a \cdot b + b \cdot a - b \cdot b \]

3. **Simplify the terms**:
We know that \(a \cdot a = |a|^2\) and \(b \cdot b = |b|^2\). Also, since the dot product is commutative, \(b \cdot a = a \cdot b\). Thus, we can rewrite the expression as:
\[ |a|^2 - a \cdot b + a \cdot b - |b|^2 \]

4. **Combine like terms**:
Notice that \(- a \cdot b + a \cdot b\) cancels out:
\[ |a|^2 - |b|^2 \]

5. **Substitute the given condition**:
Since we are given that \(|a| = |b|\), we can substitute \(|b|^2\) with \(|a|^2\):
\[ |a|^2 - |a|^2 = 0 \]

6. **Final result**:
Therefore, we find that:
\[ (a + b) \cdot (a - b) = 0 \]

### Conclusion:
The final answer is:
\[ (a + b) \cdot (a - b) = 0 \]