Question

Given vectors a, b, c such that \[{\bf{a}} \cdot ({\bf{b}} \times {\bf{c}}) \models \lambda \ne 0,\mid \] the value of \[\dfrac{{({\bf{b}} \times {\bf{c}}) \cdot ({\bf{a}} + {\bf{b}} + {\bf{c}})}}{\lambda }\] is
A. \[3\]
B. \[1\]
C. \[ - 3\lambda \]
D. \[\dfrac{3}{\lambda }\]

Answer 1

Hint:
If both magnitude and direction are present, then the quantity is called as vector. If a magnitude of on, then it is said to be unit vector. It is also known as Direction Vector. The symbol "^′' sometimes known as a cap. Such as \[\widehat a\] is used to denote Unit Vector.
It is provided by \[\widehat a = \dfrac{a}{{\left| a \right|}}\]



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



Complete step by step solution:There are two ways to define a dot product: algebraically and geometrically. The sum of the products of the matching entries of the two number sequences is the dot product, according to algebra.
We have been given in the question that,
The vectors are \[a,{\rm{ }}b,{\rm{ }}c\]
Such that,
\[{\bf{a}} \cdot ({\bf{b}} \times {\bf{c}}) = \lambda \ne 0\]--- (1)
Now, we have to determine the value of
\[\dfrac{{({\bf{b}} \times {\bf{c}}) \cdot ({\bf{a}} + {\bf{b}} + {\bf{c}})}}{\lambda }\]
For that we have to multiply the term with each term inside the parentheses, we get
\[\dfrac{{(b \times c) \cdot (a + b + c)}}{\lambda } = \dfrac{{(b \times c) \cdot a + (b \times c) \cdot b + (b \times c) \cdot c}}{\lambda }\]
Now, on solving the above obtained expression, we get
\[\dfrac{{(b \times c) \cdot a + 0 + 0}}{\lambda }\]-- (2)
Now, we have to solve the numerator of the above equation by equation (1), we have
Since: \[{\bf{a}} \cdot ({\bf{b}} \times {\bf{c}}) = \lambda \]
Now, after substituting equation (2) becomes,
\[ = \dfrac{\lambda }{\lambda }\]
Now, on simplifying the above expression, we get
\[ = 1\]
Therefore, the value of \[\dfrac{{({\bf{b}} \times {\bf{c}}) \cdot ({\bf{a}} + {\bf{b}} + {\bf{c}})}}{\lambda }\] is \[1\]



Option ‘B’ is correct

Note: Having magnitude and direction, a vector is a quantity. A unit vector is one with a magnitude of \[1\]. Cross product and dot product are two distinct concepts. We have to pay close attention while you calculate the problems. Remember that a unit vector's magnitude is always \[1\].