If `|veca|=|vecb|`, then `(veca+vecb).(veca-vecb)` is equal to

Options

A. Positive
B. Negative
C. Zero
D. None of these

Answer

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

### Step-by-Step Solution:

1. **Write down the expression:**
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) \]

2. **Apply the distributive property of the dot product:**
\[ = \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b} \]

3. **Use the property of dot product:**
- Recall that \(\vec{a} \cdot \vec{a} = |\vec{a}|^2\) and \(\vec{b} \cdot \vec{b} = |\vec{b}|^2\).
- Since \(|\vec{a}| = |\vec{b}|\), we have \(|\vec{a}|^2 = |\vec{b}|^2\).

Thus, we can rewrite the expression:
\[ = |\vec{a}|^2 - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - |\vec{b}|^2 \]

4. **Combine like terms:**
- Notice that \(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}\).
- Therefore, the expression simplifies to:
\[ = |\vec{a}|^2 - |\vec{b}|^2 \]

5. **Substituting the magnitudes:**
- Since \(|\vec{a}|^2 = |\vec{b}|^2\), we have:
\[ = |\vec{a}|^2 - |\vec{a}|^2 = 0 \]

6. **Final result:**
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b}) = 0 \]

### Conclusion:
The value of \((\vec{a} + \vec{b}) \cdot (\vec{a} - \vec{b})\) is \(0\).