Question

If vector $\overrightarrow a = \dfrac{1}{{\sqrt {10} }}(3\widehat i + \widehat k)$ and vector, $\overrightarrow b = \dfrac{1}{7}(2\widehat i + 3\widehat j - 6\widehat k)$, then find the value of $(2\overrightarrow a - \overrightarrow b ).[(\overrightarrow a \times \overrightarrow b ) \times (\overrightarrow a + 2\overrightarrow b )]$.
A. -5
B. -3
C. 5
D. 3

Answer 1

Hint: In order to solve this question first we have to expand the question with the basic cross and dot products property and then we have to convert it into most simplified equation after that we have to put the values of magnitude of vector a and b as well as their dot product in order to get the final answer.

Complete step-by-step answer:
According to the vector properties
We know that
$\overrightarrow a \times (\overrightarrow b \times \overrightarrow c ) = (\overrightarrow a .\overrightarrow c ).\overrightarrow b - (\overrightarrow a .\overrightarrow b ).\overrightarrow c $
$\overrightarrow a .\overrightarrow a = |\overrightarrow a {|^2}$
First of all we have to expand $(2\overrightarrow a - \overrightarrow b )$.$[(\overrightarrow a \times \overrightarrow b ) \times (\overrightarrow a + 2\overrightarrow b )]$ by using basic property of cross and dot product.
On expanding the given equation we have
$ = (2\overrightarrow a - \overrightarrow b ).[|\overrightarrow a {|^2}\overrightarrow b - (\overrightarrow a .\overrightarrow b ).\overrightarrow a + 2(\overrightarrow a .\overrightarrow b ).\overrightarrow b - 2|\overrightarrow b {|^2}\overrightarrow a ]$
On expanding further, we have
$ = 2(\overrightarrow a .\overrightarrow b )|\overrightarrow a {|^2} - 2|\overrightarrow a {|^2}(\overrightarrow a .\overrightarrow b ) + 4{(\overrightarrow a .\overrightarrow b )^2} - 4|\overrightarrow a {|^2}|\overrightarrow b {|^2} - |\overrightarrow a {|^2}|\overrightarrow b {|^2} + {(\overrightarrow a .\overrightarrow b )^2} + 2(\overrightarrow a .\overrightarrow b )|\overrightarrow b {|^2} - 2|\overrightarrow b {|^2}(\overrightarrow a .\overrightarrow b )$
On simplifying we get
$ = 5[(\overrightarrow a .\overrightarrow b ) - |\overrightarrow a {|^2}|\overrightarrow b {|^2}].......(1)$
In order find the value of above term, we need to find the value of $(\overrightarrow a .\overrightarrow b )$, $|\overrightarrow a |$ and $|\overrightarrow b |$.
On dot product of vector of a and b we get
$(\overrightarrow a .\overrightarrow b )$=$\dfrac{1}{{7\sqrt {10} }}(6 - 6)$
On solving we get
$(\overrightarrow a .\overrightarrow b ) = 0$
Now we will find the magnitudes of vectors a and b, for finding the square root of the sum of squares of coefficients of $\widehat i$, $\hat j$ and $\hat k$ .
Therefore we have
$|\overrightarrow a |$=$\dfrac{1}{{\sqrt {10} }}\sqrt {{3^2} + 1} $
$|\overrightarrow a | = 1$
Similarly for vector b we have
$|\overrightarrow b |$= $\dfrac{1}{7}\sqrt {{2^2} + {3^2} + {6^2}} $
$|\overrightarrow b | = 1$
 Now substituting the values of $(\overrightarrow a .\overrightarrow b )$ , $|\overrightarrow b |$ and $|\overrightarrow a |$ in equation (1) we have
$ = 5[(\overrightarrow a .\overrightarrow b ) - |\overrightarrow a {|^2}|\overrightarrow b {|^2}]$
= $5[0 - 1] \Rightarrow - 5$
Final we get
$(2\overrightarrow a - \overrightarrow b ).[(\overrightarrow a \times \overrightarrow b ) \times (\overrightarrow a + 2\overrightarrow b )] = - 5$
Hence the answer will be -5 .

Therefore the correct option is A.

Note: Don't try to solve the question by assigning the value of respective vectors. It will expand the solution because it will make it more complex and also will become quite confusing. Try to solve the equation and reduce it to its simplest form possible and then assign the respective values in order to get the answer.