Let a = 2 i - 5 j + 5 k and b = i - j + 3 k . If c is a vector such that 2 ( a × c ) + 3 ( b× c) = 0 and ( a - b ) · c = - 97 , then | c × k | 2 is equal to [2026]

Question

Let $\vec{a} = 2 \hat{i} - 5 \hat{j} + 5 \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + 3 \hat{k}$ . If $\vec{c}$ is a vector such that $2 (\vec{a} \times \vec{c}) + 3 (\vec{b} \times \vec{c}) = \vec{0}$ and $(\vec{a} - \vec{b}) \cdot \vec{c} = - 97,$ then $| \vec{c} \times \hat{k} |^{2}$ is equal to [2026]

Smart Achievers · Accepted Answer

(1)

$2 (\vec{a} \times \vec{c}) + 3 (\vec{b} \times \vec{c}) = 0$

$\Rightarrow (2 \vec{a} + 3 \vec{b}) \times \vec{c} = 0 \Rightarrow \vec{c} = λ (2 \vec{a} + 3 \vec{b})$

$\Rightarrow \vec{c} = λ (7 \hat{i} - 13 \hat{j} + 19 \hat{k})$

$Now (\vec{a} - \vec{b}) \cdot \vec{c} = λ (7 + 52 + 38) + 97 λ = - 97$

$\Rightarrow λ = - 1$

$Now \vec{c} = - 7 \hat{i} + 13 \hat{j} - 19 \hat{k}$

$\Rightarrow \vec{c} \times \hat{k} = - 7 \hat{j} + 13 \hat{i}$ $\Rightarrow$ $| \vec{c} \times \hat{k} |^{2} = 7^{2} + 13^{2} = 218$

Solution

1 218

2 205

3 193

4 233

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