Question

\( \bar{a}, \bar{b}, \bar{c} \) are three unit vectors such that \( x\bar{a} + y\bar{b} + z\bar{c} = p(\bar{b} \times \bar{c}) + q(\bar{c} \times \bar{a}) + r(\bar{a} \times \bar{b}) \). If \( (\bar{a},\bar{b})=(\bar{b},\bar{c})=(\bar{c},\bar{a})=\frac{\pi}{3} \), \( (\bar{a}, \bar{b} \times \bar{c})=\frac{\pi}{6} \) and \( \bar{a}, \bar{b}, \bar{c} \) form a right-handed system, then \( \frac{x+y+z}{p+q+r} = \)

• A. \( \frac{3}{4} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( 2\sqrt{2} \)
• D. \( \frac{3}{8} \)

Hint:

Summing symmetric equations simplifies the problem of finding sums of variables.

Answer 1

The Correct Option is D

Step 1: Calculate Box Product:

\( V = [\bar{a} \bar{b} \bar{c}] = |\bar{a}| |\bar{b} \times \bar{c}| \cos\frac{\pi}{6} = 1 \cdot \left(1 \cdot 1 \cdot \sin\frac{\pi}{3}\right) \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{3}{4} \).
Step 2: Dot Product with Basis Vectors:

Dot the main equation with \( \bar{a} \), \( \bar{b} \), \( \bar{c} \): \( \bar{a} \cdot (x\bar{a} + y\bar{b} + z\bar{c}) = pV \implies x + \frac{y}{2} + \frac{z}{2} = pV \) \( \bar{b} \cdot (x\bar{a} + y\bar{b} + z\bar{c}) = qV \implies \frac{x}{2} + y + \frac{z}{2} = qV \) \( \bar{c} \cdot (x\bar{a} + y\bar{b} + z\bar{c}) = rV \implies \frac{x}{2} + \frac{y}{2} + z = rV \)
Step 3: Sum Equations:

Summing the three equations: \( 2(x+y+z) = (p+q+r)V \) \( \frac{x+y+z}{p+q+r} = \frac{V}{2} = \frac{3/4}{2} = \frac{3}{8} \)
Step 4: Final Answer:

The ratio is \( 3/8 \).