Question

The triple product \[\left( {\vec d + \vec a} \right)\left[ {\vec a \times \left( {\vec b \times \left( {\vec c \times \vec d} \right)} \right)} \right]\] simplifies to
A.\[\left( {\vec b\vec d} \right)\left[ {\vec d\vec a\vec c} \right]\]
B.\[\left( {\vec b\vec c} \right)\left[ {\vec a\vec b\vec d} \right]\]
C.\[\left( {\vec b\vec a} \right)\left[ {\vec a\vec b\vec d} \right]\]
D.None of these

Answer 1

Hint: Here, we will rewrite the given equation using the cross product rule, \[a \times \left( {b \times c} \right) = \left( {a \cdot c} \right)b - \left( {a \cdot b} \right)c\] and then simplify it. Then we will use that if in a scalar triple product, two same variables are there, then the value of products is 0 to find the required value.

Complete step-by-step answer:
We are given that \[\left( {\vec d + \vec a} \right)\left[ {\vec a \times \left( {\vec b \times \left( {\vec c \times \vec d} \right)} \right)} \right]\].

Rewriting the given equation using the cross product rule, \[a \times \left( {b \times c} \right) = \left( {a \cdot c} \right)b - \left( {a \cdot b} \right)c\], we get
\[ \Rightarrow \left( {\vec d + \vec a} \right)\left[ {\vec a \times \left( {\left( {\vec b \cdot \vec d} \right)\vec c - \left( {\vec b \cdot \vec c} \right)\vec d} \right)} \right]\]

Simplifying the above equation by open the open brackets, we get
\[
   \Rightarrow \left( {\vec d + \vec a} \right)\left[ {\left( {\vec b \cdot \vec d} \right)\left( {\vec a \times \vec c} \right) - \left( {\vec b \cdot \vec c} \right)\left( {\vec a \times \vec d} \right)} \right] \\
   \Rightarrow \left( {\vec b \cdot \vec d} \right)\left[ {\vec d \cdot \vec a \cdot \vec c + \vec a \cdot \vec a \cdot \vec c} \right] - \left( {\vec b \cdot \vec c} \right)\left[ {\vec a \cdot \vec d \cdot \vec d + \vec a \cdot \vec a \cdot \vec d} \right] \\
 \]

We know that if in a scalar triple product, two same variables are there, then the value of products is 0.

So, the above expression becomes
\[
   \Rightarrow \left( {\vec b \cdot \vec d} \right)\left[ {\vec d \cdot \vec a \cdot \vec c + 0 \cdot \vec c} \right] - \left( {\vec b \cdot \vec c} \right)\left[ {\vec a \cdot 0 + 0 \cdot \vec d} \right] \\
   \Rightarrow \left( {\vec b \cdot \vec d} \right)\left[ {\vec d \cdot \vec a \cdot \vec c + 0} \right] - \left( {\vec b \cdot \vec c} \right)\left[ {0 + 0} \right] \\
   \Rightarrow \left( {\vec b \cdot \vec d} \right)\left[ {\vec d \cdot \vec a \cdot \vec c} \right] - \left( {\vec b \cdot \vec c} \right)\left( 0 \right) \\
   \Rightarrow \left( {\vec b \cdot \vec d} \right)\left[ {\vec d \cdot \vec a \cdot \vec c} \right] - 0 \\
   \Rightarrow \left( {\vec b \cdot \vec d} \right)\left[ {\vec d \cdot \vec a \cdot \vec c} \right] \\
 \]
Hence, option A is correct.

Note: We know that a dot product is the product between components in parallel and cross product is the product between components in perpendicular. These are because of the orthogonal direction in a product only one of the two components (parallel & perpendicular) takes part. The knowledge of both the products of vectors is really important in this question.