Let a = 2 i ^ + 7 j ^ - k ^ , b = 3 i ^ + 5 k ^ and c = i ^ - j ^ + 2 k ^ . Let d be a vector which is perpendicular to both a and b , and c · d = 12 . Then ( - i ^ + j ^ - k ^ ) · ( c × d ) is equal to [2023]

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Solution

Q.
Let $\vec{a} = 2 \hat{i} + 7 \hat{j} - \hat{k},$ $\vec{b} = 3 \hat{i} + 5 \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} + 2 \hat{k} .$ Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$ , and $\vec{c} \cdot \vec{d} = 12$ . Then $(- \hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d})$ is equal to [2023]

1 24

2 42

3 48

4 44

Ans.
(4)

$\vec{a} = 2 \hat{i} + 7 \hat{j} - \hat{k}, \vec{b} = 3 \hat{i} + 5 \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} + 2 \hat{k}$

$\vec{d} = λ (\vec{a} \times \vec{b}) = λ$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 7 & - 1 \\ 3 & 0 & 5 \end{matrix} |$ $= λ (35 \hat{i} - 13 \hat{j} - 21 \hat{k})$

$Also, \vec{c} \cdot \vec{d} = 12$

$\Rightarrow λ (35 + 13 - 42) = 12 \Rightarrow λ = 2$

$∴$ $\vec{d} = 70 \hat{i} - 26 \hat{j} - 42 \hat{k}$

$Now, (\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d})$

$= (\hat{i} + \hat{j} - \hat{k}) \cdot$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 1 & 2 \\ 70 & - 26 & - 42 \end{matrix} |$

$= (- \hat{i} + \hat{j} - \hat{k}) \cdot (94 \hat{i} + 182 \hat{j} + 44 \hat{k})$ $= - 94 + 182 - 44 = 44$

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