Question

If $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ are unit vectors such that $\bar{a} \cdot \bar{b} = \frac{1}{2}$, $\bar{c} \cdot \bar{d} = \frac{1}{2}$ and the angle between $\bar{a} \times \bar{b}$ and $\bar{c} \times \bar{d}$ is $\frac{\pi}{6}$, then the value of $|[\bar{a} \bar{b} \bar{d}] \bar{c} - [\bar{a} \bar{b} \bar{c}] \bar{d}| =$}

• A. $\frac{3}{2}$
• B. $\frac{3}{4}$
• C. $\frac{3}{8}$
• D. $2$

Hint:

Vector identities like $(\bar{A} \times \bar{B}) \times (\bar{C} \times \bar{D}) = [\bar{A}\bar{B}\bar{D}]\bar{C} - [\bar{A}\bar{B}\bar{C}]\bar{D}$ turn messy expressions into simple magnitudes.

Answer 1

The Correct Option is C

Step 1: Concept
The expression $|[\bar{a} \bar{b} \bar{d}] \bar{c} - [\bar{a} \bar{b} \bar{c}] \bar{d}|$ is the magnitude of the vector triple product $(\bar{a} \times \bar{b}) \times (\bar{c} \times \bar{d})$.

Step 2: Meaning
Use the vector triple product expansion identity: $(\bar{x}) \times (\bar{c} \times \bar{d}) = (\bar{x} \cdot \bar{d})\bar{c} - (\bar{x} \cdot \bar{c})\bar{d}$. Let $\bar{x} = \bar{a} \times \bar{b}$.

Step 3: Analysis
The magnitude is $|\bar{a} \times \bar{b}| |\bar{c} \times \bar{d}| \sin \theta$, where $\theta = \pi/6$. $|\bar{a} \times \bar{b}| = \sqrt{|\bar{a}|^2 |\bar{b}|^2 - (\bar{a} \cdot \bar{b})^2} = \sqrt{1 - 1/4} = \sqrt{3}/2$. $|\bar{c} \times \bar{d}| = \sqrt{|\bar{c}|^2 |\bar{d}|^2 - (\bar{c} \cdot \bar{d})^2} = \sqrt{1 - 1/4} = \sqrt{3}/2$. Value $= (\frac{\sqrt{3}}{2}) (\frac{\sqrt{3}}{2}) \sin(\frac{\pi}{6}) = (\frac{3}{4}) (\frac{1}{2})$.

Step 4: Conclusion
Value $= \frac{3}{8}$. Final Answer: (C)