Let , and a vector be such that and . If , then is equal to : [2025]

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Solution

Q.
Let $\vec{a} = 2 \hat{i} - 3 \hat{j} + \hat{k}$ , $\vec{b} = 3 \hat{i} + 2 \hat{j} + 5 \hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a} - \vec{c}) \times \vec{b} = - 18 \hat{i} - 3 \hat{j} + 12 \hat{k}$ and $\vec{a} \cdot \vec{c} = 3$ . If $\vec{b} \times \vec{c} = \vec{d}$ , then $| \vec{a} \cdot \vec{d} |$ is equal to : [2025]

1 12

2 18

3 15

4 9

Ans.
(3)

$\vec{a} \times \vec{b} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & - 3 & 1 \\ 3 & 2 & 5 \end{matrix} |$

$= \hat{i} (- 15 - 2) - \hat{j} (10 - 3) + \hat{k} (4 + 9) = - 17 \hat{i} - 7 \hat{j} + 13 \hat{k}$

Consider, $(\vec{a} \times \vec{b}) - ((\vec{a} - \vec{c}) \times \vec{b})$

$= (\vec{a} \times \vec{b}) - (\vec{a} \times \vec{b} - \vec{c} \times \vec{b}) = - \vec{d}$ [ $∵ \vec{b} \times \vec{c} = \vec{d}$ ]

$∴ \vec{d} = (\vec{a} - \vec{c}) \times \vec{b} - (\vec{a} \times \vec{b})$

$= - 18 \hat{i} - 3 \hat{j} + 12 \hat{k} - (- 17 \hat{i} - 7 \hat{j} + 13 \hat{k}) = - \hat{i} + 4 \hat{j} - \hat{k}$

$∴ \vec{a} \cdot \vec{d} = - 2 - 12 - 1 = - 15 \Rightarrow$ $| \vec{a} \cdot \vec{d} |$ $=$ $| - 15 |$ $= 15$ .

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