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

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. $\left[ {a{\text{ }}b{\text{ }}c} \right]d = - \left[ {c{\text{ }}b{\text{ }}a} \right]d$