Question

If $ \left[ {a \times b,b \times c,c \times a} \right] = \lambda {\left[ {abc} \right]^2} $ then $ \lambda $ is equal to
A. $ 0 $
B. $ 1 $
C. $ 2 $
D. $ 3 $

Answer 1

Hint: The question is provided in the form of a triple product. Solve the left-hand side first and then equate the two and find the value of $ \lambda $ use the properties of the triple product. $ \left( {p \times q} \right) \times \left( {r \times s} \right) = \left[ {pqs} \right]r - \left[ {pqr} \right]s $ and then $ \left[ {bcc} \right] = 0 $ at last $ \left[ {abc} \right] = \left[ {bca} \right] = \left[ {cab} \right] $ .

Complete step-by-step answer:
For solving this question, reviewing the concepts for the triple product. The representation of the triple product is $ a.\left( {b \times c} \right) = \left[ {abc} \right] $ in which $ a,b,c $ are three vectors.
Triple products are commutative in nature. That is, $ \left[ {abc} \right] = \left[ {bca} \right] = \left[ {cab} \right] $
The given statement is $ \left[ {a \times b,b \times c,c \times a} \right] = \lambda {\left[ {abc} \right]^2} $
Taking the left-hand side,
It is in the form of triple product, so can be replaced by $ \left( {a \times b} \right).\left( {\left( {b \times c} \right) \times \left( {c \times a} \right)} \right) $ $ \ldots \left( 1 \right) $
Now, the formula of cross product is
 $ \left( {p \times q} \right) \times \left( {r \times s} \right) = \left[ {pqs} \right]r - \left[ {pqr} \right]s $ here $ p,q,r,s $ represents the vectors
Now, on comparing the equation $ 1 $
 $ p = b,q = c,r = c,s = a $
writing the same formula for the inside cross product
So, $ \left( {\left( {b \times c} \right) \times \left( {c \times a} \right)} \right) = \left[ {bca} \right]c - \left[ {bcc} \right]a $
And the cross product has the property that if the vectors are the same in it then the cross product for the same is zero. So, $ \left[ {bcc} \right] = 0 $
Combing the whole equation $ 1 $
 $
  \left( {a \times b} \right).\left[ {bca} \right]c \\
   = \left( {a \times b} \right).c\left[ {bca} \right] \;
  $
Now, the first term is of the form of triple product replacing $ \left( {a \times b} \right).c $ by $ \left[ {abc} \right] $
 $ = \left[ {abc} \right]\left[ {bca} \right] $
Using the commutative property
 $ = {\left[ {abc} \right]^2} $
Equating the left-hand side and the right-hand side
 $ {\left[ {abc} \right]^2} = \lambda {\left[ {abc} \right]^2} $
Hence, $ \lambda = 1 $
So, the correct option is B.
So, the correct answer is “Option B”.

Note: In mathematics, firstly observe which formulas would take you one step closer to the solution and then proceed by taking the right steps. In this particular question use the properties of the triple product carefully. It's not like that you have to learn the steps but we can form your own unique steps.