Question

Let \( \bar{a}, \bar{b}, \bar{c} \) be three vectors each having \( \sqrt{2} \) magnitude such that

\[ (\bar{a}, \bar{b}) = (\bar{b}, \bar{c}) = (\bar{c}, \bar{a}) = \frac{\pi}{3}. \]

If

\[ \bar{x} = \bar{a} \times (\bar{b} \times \bar{c}) \quad \text{and} \quad \bar{y} = \bar{b} \times (\bar{c} \times \bar{a}), \]

then:

• A. \( |\bar{x}| = |\bar{y}| \)
• B. \( |\bar{x}| = \sqrt{2} |\bar{y}| \)
• C. (3) \( |\bar{x}| = 2 |\bar{y}| \)
• D. \( |\bar{x}| + |\bar{y}| = 2 \)

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 \( \bar{a}, \bar{b}, \bar{c} \) are given with magnitudes:

\[ |\bar{a}| = |\bar{b}| = |\bar{c}| = \sqrt{2}. \]

Also, the angle between each pair is:

\[ (\bar{a}, \bar{b}) = (\bar{b}, \bar{c}) = (\bar{c}, \bar{a}) = \frac{\pi}{3}. \]

Step 2: Vector Triple Product Expansion

Using the vector triple product identity:

\[ \bar{x} = \bar{a} \times (\bar{b} \times \bar{c}) = (\bar{a} \cdot \bar{c}) \bar{b} - (\bar{a} \cdot \bar{b}) \bar{c}. \]

Similarly,

\[ \bar{y} = \bar{b} \times (\bar{c} \times \bar{a}) = (\bar{b} \cdot \bar{a}) \bar{c} - (\bar{b} \cdot \bar{c}) \bar{a}. \]

Step 3: Magnitude of \( \bar{x} \) and \( \bar{y} \)

Since

\[ (\bar{a} \cdot \bar{b}) = (\bar{b} \cdot \bar{c}) = (\bar{c} \cdot \bar{a}) = \sqrt{2} \cdot \sqrt{2} \cos\frac{\pi}{3} = 2 \times \frac{1}{2} = 1, \]

we have:

\[ |\bar{x}|^2 = |\bar{y}|^2. \]

Taking square roots:

\[ |\bar{x}| = |\bar{y}|. \]

Step 4: Conclusion

Thus, we conclude:

\[ \boxed{|\bar{x}| = |\bar{y}|}. \]