Question

Let \( \{a} \) be a vector perpendicular to the plane containing non-zero vectors \( \{b} \) and \( \{c} \). If \( \{a}, \{b}, \{c} \) are such that \[ |\{a} + \{b} + \{c}| = \sqrt{|\{a}|^2 + |\{b}|^2 + |\{c}|^2}, \] then \[ |(\{a} \times \{b}) \cdot (\{a} \times \{c})| = \] \

• A. \( |\{a}| + |\{b}| + |\{c}| \)
• B. \( |\{a}| |\{b}| |\{c}| \)
• C. \( |\{a}|^2 + |\{b}|^2 + |\{c}|^2 \)
• D. \( |\{a}|^2 |\{c}|^2 \)
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Hint:

When dealing with perpendicular vectors, use the identity \( (\{a} \times \{b}) \cdot (\{a} \times \{c}) = |\{a}|^2 (\{b} \cdot \{c}) - (\{a} \cdot \{c}) (\{b} \cdot \{a}) \). If vectors are mutually perpendicular, this simplifies significantly.

Answer 1

The Correct Option is B

Step 1: Understanding the Given Condition
We are given: \[ |\{a} + \{b} + \{c}| = \sqrt{|\{a}|^2 + |\{b}|^2 + |\{c}|^2}. \] This equation implies that \( \{a}, \{b}, \{c} \) are mutually perpendicular vectors. That is, \[ \{a} \cdot \{b} = 0, \quad \{b} \cdot \{c} = 0, \quad \{c} \cdot \{a} = 0. \] Step 2: Evaluating \( |(\{a} \times \{b}) \cdot (\{a} \times \{c})| \)
Using the vector identity: \[ (\{a} \times \{b}) \cdot (\{a} \times \{c}) = (\{a} \cdot \{a}) (\{b} \cdot \{c}) - (\{a} \cdot \{c}) (\{b} \cdot \{a}). \] Since \( \{b} \) and \( \{c} \) are perpendicular to \( \{a} \), we know: \[ \{b} \cdot \{a} = 0, \quad \{b} \cdot \{c} = 0, \quad \{a} \cdot \{c} = 0. \] Thus, the equation simplifies to: \[ |(\{a} \times \{b}) \cdot (\{a} \times \{c})| = |\{a}| |\{b}| |\{c}|. \] Step 3: Conclusion
Since this matches option (2), the correct answer is: \[ \boxed{|\{a}| |\{b}| |\{c}|}. \] \bigskip