If `|alpha + beta| = |alpha - beta|` , then

Options

A. `alpha` is parallel to `beta`
B. `alpha` is perpendicular to `beta`
C. `|alpha| = |beta|`
D. None of the above

Answer

To solve the problem, we need to analyze the equation given:

**Given:**
\[ |\alpha + \beta| = |\alpha - \beta| \]

### Step 1: Square both sides
To eliminate the absolute values, we square both sides of the equation:

\[ |\alpha + \beta|^2 = |\alpha - \beta|^2 \]

### Step 2: Expand both sides
Using the property of the modulus, we can expand both sides:

\[ (\alpha + \beta) \cdot (\alpha + \beta) = (\alpha - \beta) \cdot (\alpha - \beta) \]

This gives us:

\[ |\alpha|^2 + 2\alpha \cdot \beta + |\beta|^2 = |\alpha|^2 - 2\alpha \cdot \beta + |\beta|^2 \]

### Step 3: Simplify the equation
Now, we can simplify the equation by canceling out the common terms on both sides:

\[ |\alpha|^2 + 2\alpha \cdot \beta + |\beta|^2 - |\alpha|^2 - |\beta|^2 = -2\alpha \cdot \beta \]

This simplifies to:

\[ 2\alpha \cdot \beta = -2\alpha \cdot \beta \]

### Step 4: Combine like terms
Now we combine like terms:

\[ 2\alpha \cdot \beta + 2\alpha \cdot \beta = 0 \]

This simplifies to:

\[ 4\alpha \cdot \beta = 0 \]

### Step 5: Conclusion
From the equation \(4\alpha \cdot \beta = 0\), we can conclude that:

\[ \alpha \cdot \beta = 0 \]

This means that the vectors \(\alpha\) and \(\beta\) are perpendicular to each other.

### Final Answer
Thus, the correct answer is that \(\alpha\) is perpendicular to \(\beta\).

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