Let , a = i ^ + j ^ - k ^ and b = 3 i ^ - j ^ + 2 k ^ . Let c be a vector such that ( a + b + c ) × c = 0 , a · c = - 17 and b · c = - 20 . Then | c × ( i ^ + j ^ + k ^ ) | 2 is [2023]

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Solution

Q.
Let $λ \in Z$ , $\vec{a} = λ \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = 3 \hat{i} - \hat{j} + 2 \hat{k}$ . Let $\vec{c}$ be a vector such that $(\vec{a} + \vec{b} + \vec{c}) \times \vec{c} = \vec{0},$ $\vec{a} \cdot \vec{c} = - 17$ and $\vec{b} \cdot \vec{c} = - 20$ . Then $| \vec{c} \times (λ \hat{i} + \hat{j} + \hat{k}) |^{2}$ is [2023]

1 46

2 53

3 49

4 62

Ans.
(1)

$As (\vec{a} + \vec{b} + \vec{c}) \times \vec{c} = 0$

$\Rightarrow (\vec{a} + \vec{b}) \times \vec{c} = 0 \Rightarrow \vec{c} = m ((λ + 3) \hat{i} + \hat{k})$

$As \vec{a} \cdot \vec{c} = - 17$

$m (λ (λ + 3) - 1) = - 17$ ...(1)

$and \vec{b} \cdot \vec{c} = - 20$

$m (3 (λ + 3) + 2) = - 20$ ...(2)

$From (1) and (2), \frac{λ^{2} + 3 λ - 1}{3 λ + 11} = \frac{17}{20}$

$20 λ^{2} + 60 λ - 20 = 51 λ + 187$

$20 λ^{2} + 9 λ - 207 = 0$

$(20 λ + 69) (λ - 3) = 0$

$\Rightarrow λ = 3 (as λ \in Z)$

$m = - 1 (from (1)) \Rightarrow \vec{c} = - (6 \hat{i} + \hat{k})$

Now, $\vec{c} \times (λ \hat{i} + \hat{j} + \hat{k}) =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 6 & 0 & - 1 \\ 3 & 1 & 1 \end{matrix} |$ $= \hat{i} + 3 \hat{j} - 6 \hat{k}$

Now, $| \vec{c} \times (λ \hat{i} + \hat{j} + \hat{k}) |^{2}$ $= 46$

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