Question

If \( \overrightarrow{a} = \alpha \hat{i} + \beta \hat{j} \) and \( \overrightarrow{b} = \alpha \hat{i} - \beta \hat{j} \) are perpendicular, where \( \alpha \neq \beta \), then \( \alpha + \beta \) is equal to:

• A. \( \alpha \beta \)
• B. \( \alpha - \beta \)
• C. \( \frac{1}{\alpha - \beta} \)
• D. \( \frac{1}{2 \alpha \beta} \)
• E. 0

Hint:

When two vectors are perpendicular, their dot product is zero. If the vectors are expressed in terms of their components, use the dot product formula to find the relationship between the components.

Answer 1

Answer: (E)

Given that \( \overrightarrow{a} = \alpha \hat{i} + \beta \hat{j} \) and \( \overrightarrow{b} = \alpha \hat{i} - \beta \hat{j} \) are perpendicular, their dot product should be zero: \[ \overrightarrow{a} \cdot \overrightarrow{b} = 0. \] Calculating the dot product: \[ (\alpha \hat{i} + \beta \hat{j}) \cdot (\alpha \hat{i} - \beta \hat{j}) = \alpha^2 - \beta^2 = 0. \] Thus, we have: \[ \alpha^2 = \beta^2. \] This implies that: \[ \alpha = \pm \beta. \] Since \( \alpha \neq \beta \), we conclude that: \[ \alpha = -\beta. \]
Therefore, \( \alpha + \beta = 0 \).
Thus, the correct answer is option (E).