Let a = 2 i ^ + 3 j ^ + 4 k ^ , b = i ^ - 2 j ^ - 2 k ^ and c = - i ^ + 4 j ^ + 3 k ^ . If d is a vector perpendicular to both b and c , and a · d = 18 , then | a × d | 2 is equal to [2023]

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Solution

Q.
Let $\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}$ , $\vec{b} = \hat{i} - 2 \hat{j} - 2 \hat{k}$ and $\vec{c} = - \hat{i} + 4 \hat{j} + 3 \hat{k}$ . If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$ , and $\vec{a} \cdot \vec{d} = 18$ , then $| \vec{a} \times \vec{d} |^{2}$ is equal to [2023]

1 720

2 640

3 680

4 760

Ans.
(1)

$∵$ $\vec{d} is perpendicular to both \vec{b} and \vec{c} .$

$∴$ $\vec{d} is parallel to plane (\vec{b} \times \vec{c}) \Rightarrow \vec{d} = λ (\vec{b} \times \vec{c})$

$Now, \vec{b} \times \vec{c} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 2 & - 2 \\ - 1 & 4 & 3 \end{matrix} |$ $= \hat{i} (- 6 + 8) - \hat{j} (3 - 2) + \hat{k} (4 - 2)$

$= 2 \hat{i} - \hat{j} + 2 \hat{k} \Rightarrow \vec{d} = λ (2 \hat{i} - \hat{j} + 2 \hat{k})$

$\Rightarrow \vec{a} \cdot \vec{d} = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) \cdot (2 λ \hat{i} - λ \hat{j} + 2 λ \hat{k})$

$\Rightarrow 18 = 4 λ - 3 λ + 8 λ \Rightarrow λ = 2$

$Thus, \vec{d} = 2 (2 \hat{i} - \hat{j} + 2 \hat{k}) = 4 \hat{i} - 2 \hat{j} + 4 \hat{k}$

$Now, \vec{a} \times \vec{d} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 4 & - 2 & 4 \end{matrix} |$ $= \hat{i} (12 + 8) - \hat{j} (8 - 16) + \hat{k} (- 4 - 12)$

$= 20 \hat{i} + 8 \hat{j} - 16 \hat{k}$ $\Rightarrow | \vec{a} \times \vec{d} |^{2} = 20^{2} + 8^{2} + {(- 16)}^{2} = 720$

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