Question

Let \( \{a}, \{b}, \{c} \) be three vectors each having \( \sqrt{2} \) magnitude such that \[ (\{a}, \{b}) = (\{b}, \{c}) = (\{c}, \{a}) = \frac{\pi}{3}. \] If \( \{x} = \{a} \times (\{b} \times \{c}) \) and \( \{y} = \{b} \times (\{c} \times \{a}) \), then \

• A. \( |\{x}| = |\{y}| \)
• B. \( |\{x}| = \sqrt{2} |\{y}| \)
• C. \( |\{x}| = 2 |\{y}| \)
• D. \( |\{x}| + |\{y}| = 2 \)
  \

Hint:

Use the vector triple product identity to simplify cross products of vector expressions. Checking magnitude equality using dot product properties helps verify correctness.

Answer 1

The Correct Option is A

Step 1: Understanding the Given Vectors
The vectors \( \{a}, \{b}, \{c} \) are given with magnitudes: \[ |\{a}| = |\{b}| = |\{c}| = \sqrt{2}. \] Also, the angle between each pair is: \[ (\{a}, \{b}) = (\{b}, \{c}) = (\{c}, \{a}) = \frac{\pi}{3}. \] Step 2: Vector Triple Product Expansion
Using the vector triple product identity: \[ \{x} = \{a} \times (\{b} \times \{c}) = (\{a} \cdot \{c}) \{b} - (\{a} \cdot \{b}) \{c}. \] Similarly, \[ \{y} = \{b} \times (\{c} \times \{a}) = (\{b} \cdot \{a}) \{c} - (\{b} \cdot \{c}) \{a}. \] Step 3: Magnitude of \( \{x} \) and \( \{y} \)
Since \( (\{a} \cdot \{b}) = (\{b} \cdot \{c}) = (\{c} \cdot \{a}) = \sqrt{2} \cdot \sqrt{2} \cos\frac{\pi}{3} = 2 \times \frac{1}{2} = 1 \), we have: \[ |\{x}|^2 = |\{y}|^2. \] Taking square roots: \[ |\{x}| = |\{y}|. \] Step 4: Conclusion
Thus, we conclude: \[ \boxed{|\{x}| = |\{y}|}. \] \bigskip