Question

If $\overrightarrow{a}$=$2\widehat{i}$+$\widehat{j}$+$2\widehat{k}$, then the value of ${{\left| \hat{i}\times (\vec{a}\times \hat{i}) \right|}^{2}}+{{\left| \hat{j}\times (\vec{a}\times \hat{j}) \right|}^{2}}+{{\left| \hat{k}\times (\vec{a}\times \hat{k}) \right|}^{2}}$is equal to ?

Answer 1

Hint: This question is of vector form. To solve this question, we first solve the cross products.Cross product of A and B with C gives you a vector that is perpendicular to C and also perpendicular to the vector A and B. By solving the cross product and putting the answers of the cross products in the given question and equating the equations , we get the value of the question .

Complete step by step solution: 
We have given the Equation $\overrightarrow{a}$=$2\widehat{i}$+$\widehat{j}$+$2\widehat{k}$,
And we have to find out the value of ${{\left| \hat{i}\times (\vec{a}\times \hat{i}) \right|}^{2}}+{{\left| \hat{j}\times (\vec{a}\times \hat{j}) \right|}^{2}}+{{\left| \hat{k}\times (\vec{a}\times \hat{k}) \right|}^{2}}$
We solve this question in the parts :-
First we take ${{\left| \hat{i}\times (\vec{a}\times \hat{i}) \right|}^{2}}$------------(1)
${{\left| \hat{i}\times (\vec{a}\times \hat{i}) \right|}^{2}}$= ${{\left| \vec{a}-(\vec{a}.\hat{i})\hat{i} \right|}^{2}}$
We put the value of $\vec{a}$in the above equation , we get
${{\left| \vec{a}-(\vec{a}.\hat{i})\hat{i} \right|}^{2}}$= ${{\left| \hat{j}+2\hat{k} \right|}^{2}}$
Solving this equation , we get
${{\left| \hat{j}+2\hat{k} \right|}^{2}}$= 1+4
= 5
Similarly we solve the equation ${{\left| \hat{j}\times (\vec{a}\times \hat{j}) \right|}^{2}}$
By putting the value of $\vec{a}$in the above equation , we get
${{\left| \hat{j}\times (\vec{a}\times \hat{j}) \right|}^{2}}$= ${{\left| 2\hat{i}+2\hat{k} \right|}^{2}}$
Solving this equation , we get
${{\left| 2\hat{i}+2\hat{k} \right|}^{2}}$= 4+4
= 8
Similarly we solve the equation \[{{\left| \hat{k}\times (\vec{a}\times \hat{k}) \right|}^{2}}\]
By putting the value of $\vec{a}$in the above equation, we get
\[{{\left| \hat{k}\times (\vec{a}\times \hat{k}) \right|}^{2}}\]= $\left| 2\hat{i}+\hat{j} \right|$
Solving this equation , we get
$\left| 2\hat{i}+\hat{j} \right|$ = 4+1
=5
Now by solving the equation (1) , (2) and (3) , we get
Value of ${{\left| \hat{i}\times (\vec{a}\times \hat{i}) \right|}^{2}}+{{\left| \hat{j}\times (\vec{a}\times \hat{j}) \right|}^{2}}+{{\left| \hat{k}\times (\vec{a}\times \hat{k}) \right|}^{2}}$
= 5 + 8 + 5
= 18
Hence, value of ${{\left| \hat{i}\times (\vec{a}\times \hat{i}) \right|}^{2}}+{{\left| \hat{j}\times (\vec{a}\times \hat{j}) \right|}^{2}}+{{\left| \hat{k}\times (\vec{a}\times \hat{k}) \right|}^{2}}$= 18
Note :-

Option (3) is correct.

Note: Students made mistake in solving the cross product of three vectors. For this, student must know the properties of cross product. It should take a lot of practice. By practicing more and more, a student can be able to solve the cross product without any confusion and it is possible to have the correct answer.