a and b are two non-zero vectors, then `( a + b). (a-b)` is equal to

Options

A. `a + b`
B. `(a-b)^(2)`
C. `(a + b)^(2)`
D. `a^(2)-b^(2)`

Answer

To solve the problem, we need to evaluate the expression \((\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b})\).

### Step-by-Step Solution:

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

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

3. **Recognize that the dot product is commutative**:
\[ \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \]
Thus, we can simplify the expression:
\[ = \mathbf{a} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b} \]

4. **Substitute the dot products with magnitudes**:
\[ = |\mathbf{a}|^2 - |\mathbf{b}|^2 \]

5. **Final result**:
\[ = |\mathbf{a}|^2 - |\mathbf{b}|^2 \]

### Conclusion:
The expression \((\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b})\) is equal to \( |\mathbf{a}|^2 - |\mathbf{b}|^2 \).