\[({\bf{a}} + {\bf{b}}) \cdot ({\bf{b}} + {\bf{c}}) \times ({\bf{a}} + {\bf{b}} + {\bf{c}}) = \]
A. \[ - \left[ {a{\rm{ }}b{\rm{ }}c} \right]\]
B. \[\left[ {a{\rm{ }}b{\rm{ }}c{\rm{ }}} \right]\]
C. \[0\]
D. \[2[abc]\]

Hint: Vectors can be multiplied in two main methods: scalar product or dot product, which generates a scalar, and vector product or cross product, which produces a vector. The scalar product of the two provided vectors is represented by the dot product of two vectors. In this case\[({\bf{a}} + {\bf{b}}) \cdot ({\bf{b}} + {\bf{c}}) \times ({\bf{a}} + {\bf{b}} + {\bf{c}})\], we have to multiply the term \[\left( {b + c} \right)\]with the one inside the parentheses, by doing so; we can obtain desired result after simplifying the resultant expression.



Formula Used:The dot product of two vectors can be calculated using the below formula,
\[{\bf{a}}.\left( {{\bf{b}}{\rm{ }} + {\rm{ }}{\bf{c}}} \right){\rm{ }} = {\rm{ }}{\bf{a}}.{\bf{b}}{\rm{ }} + {\rm{ }}{\bf{a}}.{\bf{c}}\]



Complete step by step solution:We have been given in the question that,
\[(a + b).(b + c) \times (a + b + c)\]
Now, we have to solve the given expression to obtain the value.
For that we have to determine that using dot product of vectors.
Let’s multiply the term \[\left( {b + c} \right)\] with each term inside the parentheses by expanding, we get
\[ = (a + b) \cdot \{ - a \times b + c \times a + b \times b + c \times b + b \times c + c \times c\} \]
Now, we have to simplify the terms inside the parentheses.
Since \[b \times b = 0\] and \[c \times c = 0\], we obtain
\[ = (a + b).\{ - a \times b + b \times c + c \times a + c \times b\} \]
After simplification, now the equation becomes,
\[ = (a + b).( - a \times b + c \times a)\]
Now, we have to multiply the term \[(a + b)\] with \[( - a \times b + c \times a)\] we get
\[ = - [aab] + [aca] - [bab] + [bca]\]
On simplifying the obtained expression, we get
\[ = [abc]\]
Therefore, the value of \[({\bf{a}} + {\bf{b}}) \cdot ({\bf{b}} + {\bf{c}}) \times ({\bf{a}} + {\bf{b}} + {\bf{c}})\] is \[[abc]\]



Option ‘B’ is correct

Note: The dot product notion claims that any two vectors can be multiplied to yield the scalar quantity. It's used to get the merchandise. It returns the products of two vectors or more vectors in two or more dimensions. So, students should be thorough with the concepts of dot product and cross product to solve these types of problems and mistakes made in these types of problems while doing multiplication with different signs and with the brackets.