Let R , a= i+2 j-3 k, b=i- j+2 k If ((a+b) (a b)) (a-b)=8 i-40 j-… - Vidyadip

MCQ

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 $((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b})=8 \hat{i}-40 \hat{j}-24 \hat{k}$, then $|\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|^2$ is equal to

✓
$140$
B
$132$
C
$144$
D
$136$

✓

Answer

Correct option: A.

$140$

a
$\overrightarrow{ a }=\lambda \hat{ i }+2 \hat{ j }-3 \hat{ k }$

$\overrightarrow{ b }=\hat{ i }-\lambda \hat{ j }+2 \hat{ k }$ $\Rightarrow(\overrightarrow{ b }-\overrightarrow{ a }) \times((\overrightarrow{ a }+\overrightarrow{ b }) \times(\overrightarrow{ a } \times \overrightarrow{ b }))=8 \hat{ i }-40 \hat{ j }-24 \hat{ k }$

$\Rightarrow((\overrightarrow{ a }-\overrightarrow{ b }) \cdot(\overrightarrow{ a }+\overrightarrow{ b }))(\overrightarrow{ a } \times \overrightarrow{ b })=8 \hat{ i }-40 j -24 \hat{ k }$

$\Rightarrow 8(\overrightarrow{ a } \times \overrightarrow{ b })=8 \hat{ i }-40 \hat{ j }-24 \hat{ k }$

$\text { Now, } \overrightarrow{ a } \times \overrightarrow{ b }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ \lambda & 2 & -3 \\ 1 & -\lambda & 2\end{array}\right|$

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

$\Rightarrow \lambda=1$

$\therefore \overrightarrow{ a }=\hat{ i }+2 \hat{ j }-3 \hat{ k }$

$\overrightarrow{ b }=\hat{ i }-\hat{ j }+2 \hat{ k }$

$\Rightarrow \lambda=1$

$\therefore \overrightarrow{ a }=\hat{ i }+2 \hat{ j }-3 \hat{ k }$

$\overrightarrow{ b }=\hat{ i }-\hat{ j }+2 \hat{ k }$

$\Rightarrow \overrightarrow{ a }+\overrightarrow{ b }=2 \hat{ i }+\hat{ j }-\hat{ k }, \overrightarrow{ a }-\overrightarrow{ b }=3 \hat{ j }-5 \hat{ k }$ $\Rightarrow(\overrightarrow{ a }+\overrightarrow{ b }) \times(\overrightarrow{ a }-\overrightarrow{ b })=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 1 & -1 \\ 0 & 3 & -5\end{array}\right|=2$

$\hat{ i }+10 \hat{ j }+6 \hat{ k }$

$\therefore \text { required answer }=4+100+36=140$

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