Question

If \( a=|\bar{a}| \); \( b=|\bar{b}| \) then \( \left(\frac{\bar{a}}{a^2} - \frac{\bar{b}}{b^2}\right)^2 = \)

• A. \( \left(\frac{\bar{a}-\bar{b}}{a^2b^2}\right)^2 \)
• B. \( \left(\frac{\bar{a}-\bar{b}}{ab}\right)^2 \)
• C. \( \left(\frac{b\bar{a}-a\bar{b}}{ab}\right)^2 \)
• D. \( \left(\frac{a\bar{a}-b\bar{b}}{a^2b^2}\right)^2 \)

Hint:

Use \( |\vec{v}|^2 = \vec{v} \cdot \vec{v} \) to expand.

Answer 1

The Correct Option is B

Step 1: Expand the Square:

\[ \left( \frac{\bar{a}}{a^2} - \frac{\bar{b}}{b^2} \right)^2 = \frac{|\bar{a}|^2}{a^4} + \frac{|\bar{b}|^2}{b^4} - \frac{2 \bar{a} \cdot \bar{b}}{a^2 b^2} \] \[ = \frac{1}{a^2} + \frac{1}{b^2} - \frac{2 \bar{a} \cdot \bar{b}}{a^2 b^2} \] \[ = \frac{b^2 + a^2 - 2 \bar{a} \cdot \bar{b}}{a^2 b^2} \] \[ = \frac{|\bar{a} - \bar{b}|^2}{(ab)^2} = \left( \frac{|\bar{a} - \bar{b}|}{ab} \right)^2 \]
Step 2: Final Answer:

Option (B).