Let a ? = 2 i ^ – 3 j ^ + 4 k ^ , b ? = 3 i ^ + 4 j ^ – 5 k ^ and a vector c ? be such that a ? × ( b ? + c ? ) + b ? × c ? = i ^ + 8 j ^ + 13 k ^ . If a ? · c ? = 13 , then ( 24 – b ? · c ? ) is equal to [2024]

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Solution

Q.
Let $\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ , $\vec{b} = 3 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and a vector $\vec{c}$ be such that $\vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8 \hat{j} + 13 \hat{k}$ . If $\vec{a} \cdot \vec{c} = 13$ , then $(24 - \vec{b} \cdot \vec{c})$ is equal to __________. [2024]

Ans.
(46)

We have, $\vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times \vec{c} = \hat{i} + 8 \hat{j} + 13 \hat{k}$

$\Rightarrow (\vec{a} \times \vec{b}) + (\vec{a} \times \vec{c}) + (\vec{b} \times \vec{c}) = \hat{i} + 8 \hat{j} + 13 \hat{k}$

$\Rightarrow \vec{a} \times (\vec{a} \times \vec{b}) + \vec{a} \times (\vec{a} \times \vec{c}) + \vec{a} \times (\vec{b} \times \vec{c}) = \vec{a} \times (\hat{i} + 8 \hat{j} + 13 \hat{k})$

$\Rightarrow (\vec{a} \cdot \vec{b}) \vec{a} - a^{2} \vec{b} + (\vec{a} \cdot \vec{c}) \vec{a} - a^{2} \vec{c} + (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} = \vec{a} \times (\hat{i} + 8 \hat{j} + 13 \hat{k})$

$\Rightarrow - 26 \vec{a} - 29 \vec{b} + 13 \vec{a} - 29 \vec{c} + 13 \vec{b} + 26 \vec{c} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & - 3 & 4 \\ 1 & 8 & 13 \end{matrix} |$

$\Rightarrow - 13 \vec{a} - 16 \vec{b} - 3 \vec{c} = - 71 \hat{i} - 22 \hat{j} + 19 \hat{k}$

$\Rightarrow - 13 \vec{a} \cdot \vec{b} - 16 b^{2} - 3 \vec{b} \cdot \vec{c} = - 213 - 88 - 95$

$\Rightarrow 338 - 800 - 3 \vec{b} \cdot \vec{c} = - 396$

$\Rightarrow - 462 - 3 \vec{b} \cdot \vec{c} = - 396 \Rightarrow \vec{b} \cdot \vec{c} = - 22$

. $∴ 24 - \vec{b} \cdot \vec{c} = 24 + 22 = 46$

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