Question

If \[a,b,c,d\] are four vectors then the value of \[\left( {a \times b} \right) \times \left( {c \times d} \right) + \left( {a \times c} \right) \times \left( {d \times b} \right) + \left( {a \times d} \right) \times \left( {b \times c} \right)\] is ___________

Answer 1

Hint: In this problem, use the two formulas of the vector product of four vectors and then convert and simplify further by cancelation of the common terms to get the required value of the given problem.

Complete step-by-step answer:
Given that \[a,b,c,d\] are four vectors.
We have to find the value of \[\left( {a \times b} \right) \times \left( {c \times d} \right) + \left( {a \times c} \right) \times \left( {d \times b} \right) + \left( {a \times d} \right) \times \left( {b \times c} \right)\]
We know that \[\left( {a \times b} \right) \times \left( {c \times d} \right) = \left[ {a{\text{ }}b{\text{ }}d} \right]c - \left[ {a{\text{ }}b{\text{ }}c} \right]d\]
Let \[v = \left( {a \times b} \right) \times \left( {c \times d} \right) + \left( {a \times c} \right) \times \left( {d \times b} \right) + \left( {a \times d} \right) \times \left( {b \times c} \right)\]. By using the above formula, we have
\[ \Rightarrow v = \left[ {a{\text{ }}b{\text{ }}d} \right]c - \left[ {a{\text{ }}b{\text{ }}c} \right]d + \left( {a \times c} \right) \times \left( {d \times b} \right) + \left( {a \times d} \right) \times \left( {b \times c} \right){\text{ }}\left[ {\because \left( {a \times b} \right) \times \left( {c \times d} \right) = \left[ {a{\text{ }}b{\text{ }}d} \right]c - \left[ {a{\text{ }}b{\text{ }}c} \right]d} \right]\]
Also, we know that \[\left( {a \times b} \right) \times \left( {c \times d} \right) = \left[ {a{\text{ }}c{\text{ }}d} \right]b - \left[ {b{\text{ }}c{\text{ }}d} \right]a\]
\[ \Rightarrow v = \left[ {a{\text{ }}b{\text{ }}d} \right]c - \left[ {a{\text{ }}b{\text{ }}c} \right]d + \left[ {a{\text{ }}d{\text{ }}b} \right]c - \left[ {c{\text{ }}d{\text{ }}b} \right]a + \left( {a \times d} \right) \times \left( {b \times c} \right){\text{ }}\left[ {\because \left( {a \times b} \right) \times \left( {c \times d} \right) = \left[ {a{\text{ }}c{\text{ }}d} \right]b - \left[ {b{\text{ }}c{\text{ }}d} \right]a} \right]\]
Again, using the formula \[\left( {a \times b} \right) \times \left( {c \times d} \right) = \left[ {a{\text{ }}c{\text{ }}d} \right]b - \left[ {b{\text{ }}c{\text{ }}d} \right]a\] we get
\[ \Rightarrow v = \left[ {a{\text{ }}b{\text{ }}d} \right]c - \left[ {a{\text{ }}b{\text{ }}c} \right]d + \left[ {a{\text{ }}d{\text{ }}b} \right]c - \left[ {c{\text{ }}d{\text{ }}b} \right]a + \left[ {a{\text{ }}b{\text{ }}c} \right]d - \left[ {d{\text{ }}b{\text{ }}c} \right]a{\text{ }}\left[ {\because \left( {a \times b} \right) \times \left( {c \times d} \right) = \left[ {a{\text{ }}c{\text{ }}d} \right]b - \left[ {b{\text{ }}c{\text{ }}d} \right]a} \right]\]
Cancelling the common terms, we have
\[
   \Rightarrow v = \left[ {a{\text{ }}b{\text{ }}d} \right]c + \left[ {a{\text{ }}d{\text{ }}b} \right]c - \left[ {c{\text{ }}d{\text{ }}b} \right]a - \left[ {d{\text{ }}b{\text{ }}c} \right]a \\
   \Rightarrow v = \left[ {a{\text{ }}b{\text{ }}d} \right]c - \left[ {a{\text{ }}b{\text{ }}d} \right]c - \left[ {c{\text{ }}d{\text{ }}b} \right]a - \left[ {d{\text{ }}b{\text{ }}c} \right]a{\text{ }}\left[ {\because \left[ {a{\text{ }}b{\text{ }}c} \right]d = - \left[ {a{\text{ }}c{\text{ }}b} \right]d} \right] \\
   \Rightarrow v = - \left[ {c{\text{ }}d{\text{ }}b} \right]a - \left[ {d{\text{ }}b{\text{ }}c} \right]a \\
   \Rightarrow v = - \left[ {c{\text{ }}d{\text{ }}b} \right]a - \left[ {c{\text{ }}d{\text{ }}b} \right]a{\text{ }}\,{\text{ }}\left[ {\because \left[ {a{\text{ }}b{\text{ }}c} \right]d = \left[ {c{\text{ }}a{\text{ }}b} \right]d} \right] \\
   \Rightarrow v = - 2\left[ {c{\text{ }}d{\text{ }}b} \right]a \\
  \therefore v = 2\left[ {b{\text{ }}c{\text{ }}d} \right]a{\text{ }}\left[ {\because \left[ {a{\text{ }}b{\text{ }}c} \right]d = - \left[ {c{\text{ }}b{\text{ }}a} \right]d} \right] \\
\]
Thus, the value of \[\left( {a \times b} \right) \times \left( {c \times d} \right) + \left( {a \times c} \right) \times \left( {d \times b} \right) + \left( {a \times d} \right) \times \left( {b \times c} \right)\] is \[2\left[ {b{\text{ }}c{\text{ }}d} \right]a\].

Note: Here we have to remember the formulae
           1. \[\left( {a \times b} \right) \times \left( {c \times d} \right) = \left[ {a{\text{ }}c{\text{ }}d} \right]b - \left[ {b{\text{ }}c{\text{ }}d} \right]a\]
           2. \[\left( {a \times b} \right) \times \left( {c \times d} \right) = \left[ {a{\text{ }}b{\text{ }}d} \right]c - \left[ {a{\text{ }}b{\text{ }}c} \right]d\]
           3. \[\left[ {a{\text{ }}b{\text{ }}c} \right]d = - \left[ {a{\text{ }}c{\text{ }}b} \right]d\]
           4. \[\left[ {a{\text{ }}b{\text{ }}c} \right]d = - \left[ {c{\text{ }}b{\text{ }}a} \right]d\]