`(a +b). (a-b) =0` implies that

Options

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

Answer

To solve the equation \((\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = 0\), we can follow these steps:

### Step 1: Expand the Left-Hand Side
Using the distributive property of the dot product, we can expand the expression:
\[ (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} - \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b} \]

### Step 2: Simplify the Expression
Since the dot product is commutative, \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\). Thus, we can rewrite the expression:
\[ \mathbf{a} \cdot \mathbf{a} - \mathbf{b} \cdot \mathbf{b} = 0 \]

### Step 3: Rearrange the Equation
Rearranging the equation gives us:
\[ \mathbf{a} \cdot \mathbf{a} = \mathbf{b} \cdot \mathbf{b} \]

### Step 4: Interpret the Result
The expression \(\mathbf{a} \cdot \mathbf{a}\) represents the square of the magnitude of vector \(\mathbf{a}\) (i.e., \(|\mathbf{a}|^2\)), and \(\mathbf{b} \cdot \mathbf{b}\) represents the square of the magnitude of vector \(\mathbf{b}\) (i.e., \(|\mathbf{b}|^2\)). Therefore, we have:
\[ |\mathbf{a}|^2 = |\mathbf{b}|^2 \]

### Step 5: Conclusion
This implies that the magnitudes of vectors \(\mathbf{a}\) and \(\mathbf{b}\) are equal:
\[ |\mathbf{a}| = |\mathbf{b}| \]

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