Let R,a= i+2j-3k,b=i- j+2k.

Question

Let $\lambda \in R,\vec{a}=\lambda \hat{i}+2\hat{j}-3\hat{k},\vec{b}=\hat{i}-\lambda \hat{j}+2\hat{k}$.

If $((\vec{a}+\vec{b})	imes (\vec{a}	imes \vec{b}))	imes (\vec{a}-\vec{b})=8\hat{i}-40\hat{j}-24\hat{k}$,

then $|\lambda (\vec{a}+\vec{b})	imes (\vec{a}-\vec{b}){|}^{2}$ is equal to :

Accepted Answer

$(\overset{	o }{a}	imes (\overset{	o }{a}	imes \overset{	o }{b})+\overset{	o }{b}	imes (\overset{	o }{a}	imes \overset{	o }{b}))	imes (\overset{	o }{a}-\overset{	o }{b})$

$=(\overset{	o }{a}(\overset{	o }{a}.\, \overset{	o }{b})-\overset{	o }{b}(\overset{	o }{a}.\, \overset{	o }{a})+\overset{	o }{a}(\overset{	o }{b}.\, \overset{	o }{b})-\overset{	o }{b}(\overset{	o }{a}.\, \overset{	o }{b}))	imes (\overset{	o }{a}-\overset{	o }{b})$

$=(\overset{	o }{a}.\, \overset{	o }{b})(\overset{	o }{a}	imes \overset{	o }{a}-\overset{	o }{a}	imes \overset{	o }{b})-(\overset{	o }{a}.\, \overset{	o }{a})(\overset{	o }{b}	imes \overset{	o }{a}-\overset{	o }{b}	imes \overset{	o }{b})+(\overset{	o }{b}.\, \overset{	o }{b})(\overset{	o }{a}	imes \overset{	o }{a}-\overset{	o }{a}	imes \overset{	o }{b})-(\overset{	o }{a}.\, \overset{	o }{b})(\overset{	o }{b}	imes \overset{	o }{a}-\overset{	o }{b}	imes \overset{	o }{b})$

$=(\overset{	o }{a}.\, \overset{	o }{b})(\overset{	o }{b}	imes \overset{	o }{a})-(\overset{	o }{a}.\, \overset{	o }{a})(\overset{	o }{b}	imes \overset{	o }{a})+(\overset{	o }{b}.\, \overset{	o }{b})(\overset{	o }{b}	imes \overset{	o }{a})-(\overset{	o }{a}.\, \overset{	o }{b})(\overset{	o }{b}	imes \overset{	o }{a})$

$=(\overset{	o }{b}	imes \overset{	o }{a})(\overset{	o }{b}.\, \overset{	o }{b}-\overset{	o }{a}.\, \overset{	o }{a})$

$=(5\overset{	o }{b}	imes \overset{	o }{a})(5+{\lambda }^{2}-13-{\lambda }^{2})$

$=8(\overset{	o }{a}	imes \overset{	o }{b})$

$\overset{	o }{a}	imes \overset{	o }{b}=\begin{vmatrix} \hat{i} \quad  \hat{j} \quad  \hat{k} \ \lambda  \quad  2 \quad  -3 \ 1 \quad  -\lambda  \quad  2 \end{vmatrix}$

$=\hat{i}(4-3\lambda )-\hat{j}(2\lambda +3)+\hat{k}(-{\lambda }^{2}-2)$

$\Rightarrow \lambda =1$

${|\overset{	o }{a}	imes (\overset{	o }{a}-\overset{	o }{b})+\overset{	o }{b}	imes (\overset{	o }{a}-\overset{	o }{b})|}^{2}$

$={|2(\overset{	o }{a}	imes \overset{	o }{b})|}^{2}=4\, .\, 35=140$

Solutions · 1