Question

If \(\vec c\) and \[ (\vec b\times \vec c)\times(\vec c\times \vec a) \] are parallel vectors, then \[ \left[\vec c\times\vec a\;\;\; \vec a\times\vec b\;\;\; \vec b\times\vec c\right] = \]

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

Hint:

Remember the important identity \( [\vec a\times\vec b,\vec b\times\vec c,\vec c\times\vec a] = [\vec a,\vec b,\vec c]^2 \). It frequently appears in advanced vector algebra MCQs.

Answer 1

The Correct Option is C

Concept: Use the vector identity \[ (\vec p\times\vec q)\times(\vec r\times\vec s) = [\vec p\;\vec q\;\vec s]\vec r - [\vec p\;\vec q\;\vec r]\vec s. \] Also, the scalar triple product is invariant under cyclic permutations.

Step 1: Evaluate \((\vec b\times\vec c)\times(\vec c\times\vec a)\). Using the identity, \[ (\vec b\times\vec c)\times(\vec c\times\vec a) = [\vec b\;\vec c\;\vec a]\vec c - [\vec b\;\vec c\;\vec c]\vec a. \] Since \[ [\vec b\;\vec c\;\vec c]=0, \] we get \[ (\vec b\times\vec c)\times(\vec c\times\vec a) = [\vec b\;\vec c\;\vec a]\vec c. \] Thus the vector is automatically parallel to \(\vec c\).

Step 2: Find its magnitude. Taking magnitudes, \[ \left|(\vec b\times\vec c)\times(\vec c\times\vec a)\right| = \left|[\vec b\;\vec c\;\vec a]\right|\,|\vec c|. \] Therefore, \[ \left|[\vec b\;\vec c\;\vec a]\right| = \frac{\left|(\vec b\times\vec c)\times(\vec c\times\vec a)\right|} {|\vec c|}. \]

Step 3: Evaluate the required scalar triple product. Using the identity \[ [\vec c\times\vec a,\;\vec a\times\vec b,\;\vec b\times\vec c] = [\vec a,\vec b,\vec c]^2, \] we obtain \[ [\vec c\times\vec a,\;\vec a\times\vec b,\;\vec b\times\vec c] = \left([\vec b\;\vec c\;\vec a]\right)^2. \] Substituting the result from Step 2, \[ [\vec c\times\vec a,\;\vec a\times\vec b,\;\vec b\times\vec c] = \left( \frac{\left|(\vec b\times\vec c)\times(\vec c\times\vec a)\right|} {|\vec c|} \right)^2. \] Therefore, \[ \boxed{ [\vec c\times\vec a,\;\vec a\times\vec b,\;\vec b\times\vec c] = \left( \frac{\left|(\vec b\times\vec c)\times(\vec c\times\vec a)\right|} {|\vec c|} \right)^2 } \] \[ \boxed{\text{Answer = (C)}} \]