Question

The value of \[\vec a \cdot ((\vec b \times \vec c) \times (\vec a + \vec b + \vec c))\] is equal to
A. \[[abc]\]
B. \[2[abc]\]
C. \[3[abc]\]
D. 0

Answer 1

Hint: To answer this question, first rewrite it as \[ = {\bf{A}} \cdot \{ {\bf{B}} \times {\bf{A}} + {\bf{B}} \times {\bf{B}} + {\bf{B}} \times {\bf{C}} + {\bf{C}} \times {\bf{A}} + {\bf{C}} \times {\bf{B}} + {\bf{C}} \times {\bf{C}}\} \] using dot products and cross products, and then simplify it by doing cross products and opening the brackets.

Formula Used: The formula for dot product is given by,
\[{\bf{a}}.\left( {{\bf{b}}{\rm{ }} + {\rm{ }}{\bf{c}}} \right){\rm{ }} = {\rm{ }}{\bf{a}}.{\bf{b}}{\rm{ }} + {\rm{ }}{\bf{a}}.{\bf{c}}\]

Complete step by step solution:
 We have been given in the question that
\[\vec a \cdot ((\vec b \times \vec c) \times (\vec a + \vec b + \vec c))\]
\[{\bf{A}} \cdot \{ ({\bf{B}} + {\bf{C}}) \times ({\bf{A}} + {\bf{B}} + {\bf{C}})\} \]
Now, we have to multiply the term each term with the other term inside the parentheses, we get
\[ = a \cdot [(b \times a + b \times b + b \times c + c \times a + c \times b + c \times c)]\]
Now, let us simplify the above expression, we get
\[ = a \cdot [(b \times a) + (b \times c) + (c \times a) + (c \times b)]\]
Now, we have to multiply \[{\bf{A}}\]with each term inside the parentheses, we obtain
We further reduce this to be able to solve it by doing the cross product of the two vectors beginning with the bracket.
\[ = a[(b \times a) + (b \times c) + (c \times a) - (b \times c)]\]
Now, let’s simplify the terms to make it less complicated, we get
\[ = a[(b \times a) + (c \times a)]\]
Multiply the terms inside the parentheses with one in the outside, we get
\[ = [aba] + [aca]\]
Thus, we obtain
\[ = 0 + 0 = 0\]
For any non-zero vector \[a,b,c\]
\[a \cdot ((b + c) \times (a + b + c)] = 0\]
Therefore, the value of \[\vec a \cdot ((\vec b \times \vec c) \times (\vec a + \vec b + \vec c))\] is equal to \[0\]

Option ‘D’ is correct
Note: Students should be clear with the concepts of scalar and vectors and also the dot products of vectors and cross products of vectors. Let us assume there are three vectors, say, a, b, and c. The vector triple product of a, b, and c is the cross-product of vectors such as a (b c) and (a b) c. As a result, it can be expressed as \[a \times \left( {b \times c} \right) = \left( {a.{\rm{ }}c} \right)b - \left( {a.{\rm{ }}b} \right){\rm{ }}c\]. The two vectors enclosed in brackets are combined linearly to form the vector triple product \[a{\rm{ }}b{\rm{ }}c\].