Let R , a = i ^ + 2 j ^ - 3 k ^ , b = i ^ - j ^ + 2 k ^ . If ( ( a + b ) × ( a × b ) ) × ( a - b ) = 8 i ^ - 40 j ^ - 24 k ^ , then | ( a + b ) × ( a - b ) | 2 is equal to [2023]

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Solution

Q.
Let $λ \in R,$ $\vec{a} = λ \hat{i} + 2 \hat{j} - 3 \hat{k},$ $\vec{b} = \hat{i} - λ \hat{j} + 2 \hat{k} .$ 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 $| λ (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) |^{2}$ is equal to [2023]

1 136

2 132

3 140

4 144

Ans.
(3)

$Given \vec{a} = λ \hat{i} + 2 \hat{j} - 3 \hat{k}$ and $\vec{b} = \hat{i} - λ \hat{j} + 2 \hat{k}$

$Let \vec{u} = \vec{a} + \vec{b} = (λ + 1) \hat{i} + (2 - λ) \hat{j} - \hat{k}$

$\vec{v} = \vec{a} \times \vec{b} = (4 - 3 λ) \hat{i} - (2 λ + 3) \hat{j} - (λ^{2} + 2) \hat{k}$

$\vec{w} = \vec{a} - \vec{b} = (λ - 1) \hat{i} + (2 + λ) \hat{j} - 5 \hat{k}$

$(\vec{u} \times \vec{v}) \times \vec{w} = (32 - 24 λ) \hat{i} + (- 16 λ - 24) \hat{j} + (- 8 λ^{2} - 16) \hat{k}$

$= 8 \hat{i} - 40 \hat{j} - 24 \hat{k}$

By comparing we get, $- 16 λ - 24 = - 40 \Rightarrow - 16 λ = - 16 \Rightarrow λ = 1$

$∴$ $\vec{a} + \vec{b} = 2 \hat{i} + \hat{j} - \hat{k} and \vec{a} - \vec{b} = 3 \hat{j} - 5 \hat{k}$

$Now, (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & - 1 \\ 0 & 3 & - 5 \end{matrix} |$

$= (- 5 + 3) \hat{i} - \hat{j} (- 10) + \hat{k} (6) = - 2 \hat{i} + 10 \hat{j} + 6 \hat{k}$

$| (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) | = \sqrt{4 + 100 + 36} = \sqrt{140}$

$∴$ $| λ (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) |^{2} = 140$

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