Question:
If \(\bar{a},\bar{b},\bar{c}\) are vectors such that \([\bar{a}\ \bar{b}\ \bar{c}]=3\), then \([\bar{a}\times \bar{b}\ \ \bar{b}\times \bar{c}\ \ \bar{c}\times \bar{a}]=\)
If \(\bar{a},\bar{b},\bar{c}\) are vectors such that \([\bar{a}\ \bar{b}\ \bar{c}]=3\), then \([\bar{a}\times \bar{b}\ \ \bar{b}\times \bar{c}\ \ \bar{c}\times \bar{a}]=\)
Show Hint
Use the identity that the triple product of the three cross products equals the square of the original triple product.
Updated On: Nov 12, 2026
- \(6\)
- \(9\)
- \(12\)
- \(15\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
The standard identity is \([\bar{a}\times\bar{b}\ \ \bar{b}\times\bar{c}\ \ \bar{c}\times\bar{a}]=[\bar{a}\ \bar{b}\ \bar{c}]^{2}\).
Step 2: Where it comes from
Expand \((\bar{b}\times\bar{c})\times(\bar{c}\times\bar{a})=[\bar{b}\ \bar{c}\ \bar{a}]\,\bar{c}\). Then dot with \(\bar{a}\times\bar{b}\) to get \([\bar{a}\ \bar{b}\ \bar{c}]\cdot[\bar{a}\ \bar{b}\ \bar{c}]\).
Step 3: Substitute
With \([\bar{a}\ \bar{b}\ \bar{c}]=3\), the value is \(3^{2}=9\).
Final Answer:
The value is \(9\). \[ \boxed{\text{(B) }9} \]
The standard identity is \([\bar{a}\times\bar{b}\ \ \bar{b}\times\bar{c}\ \ \bar{c}\times\bar{a}]=[\bar{a}\ \bar{b}\ \bar{c}]^{2}\).
Step 2: Where it comes from
Expand \((\bar{b}\times\bar{c})\times(\bar{c}\times\bar{a})=[\bar{b}\ \bar{c}\ \bar{a}]\,\bar{c}\). Then dot with \(\bar{a}\times\bar{b}\) to get \([\bar{a}\ \bar{b}\ \bar{c}]\cdot[\bar{a}\ \bar{b}\ \bar{c}]\).
Step 3: Substitute
With \([\bar{a}\ \bar{b}\ \bar{c}]=3\), the value is \(3^{2}=9\).
Final Answer:
The value is \(9\). \[ \boxed{\text{(B) }9} \]
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