Question

For any non-zero vectors \( \vec{a} \) and \( \vec{b} \), \( \left| \vec{b} \times (\vec{a} \times \vec{b}) \right| = \)

• A. \( \left| \vec{a} \times \vec{b} \right| \)
• B. \( \left| \vec{a} \times \vec{b} \right|^2 \)
• C. 0
• D. \( \vec{a} \times \vec{b} \)

Hint:

For vector triple products, the identity \( \vec{b} \times (\vec{a} \times \vec{b}) = (\vec{b} \cdot \vec{b}) \vec{a} - (\vec{b} \cdot \vec{a}) \vec{b} \) is very useful in simplifying expressions.

Answer 1

The Correct Option is B

Step 1: Understanding the expression.
We are asked to evaluate the expression \( \left| \vec{b} \times (\vec{a} \times \vec{b}) \right| \). The vector triple product identity gives us the following result: \[ \vec{b} \times (\vec{a} \times \vec{b}) = (\vec{b} \cdot \vec{b}) \vec{a} - (\vec{b} \cdot \vec{a}) \vec{b} \] This simplifies to: \[ \vec{b} \times (\vec{a} \times \vec{b}) = |\vec{b}|^2 \vec{a} - (\vec{b} \cdot \vec{a}) \vec{b} \]
Step 2: Taking the magnitude.
Now, taking the magnitude of both sides, we get: \[ \left| \vec{b} \times (\vec{a} \times \vec{b}) \right| = \left| |\vec{b}|^2 \vec{a} - (\vec{b} \cdot \vec{a}) \vec{b} \right| \] For simplicity, we focus on the fact that the magnitude of the cross product is related to the area of the parallelogram formed by the vectors. Therefore, the magnitude of \( \vec{b} \times (\vec{a} \times \vec{b}) \) is equal to the square of the magnitude of \( \vec{a} \times \vec{b} \). Hence, the correct answer is \( \left| \vec{a} \times \vec{b} \right|^2 \).
Step 3: Conclusion.
Thus, the correct answer is \( \left| \vec{a} \times \vec{b} \right|^2 \), corresponding to option (B).