Question
Let $\overrightarrow{a} = 2\widehat{i} - 5\widehat{j} + 5k$ and $\overrightarrow{b} = \widehat{i} - \widehat{j} + 3k$. If $\overrightarrow{c}$ is a vector such that
$2(\overrightarrow{a} \times \overrightarrow{c}) + 3(\overrightarrow{b} \times \overrightarrow{c}) = \overrightarrow{0}$ and $(\overrightarrow{a} - \overrightarrow{b}) \cdot \overrightarrow{c} = - 97$, then $|\overrightarrow{c} \times k|^{2}$ is equal to
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Solution
$2(\overrightarrow{a} \times \overrightarrow{c}) + 3(\overrightarrow{b} \times \overrightarrow{c}) = 0$
$${\Rightarrow (2\overrightarrow{a} + 3\overrightarrow{\text{ }d}) \times \overrightarrow{c} = 0 \Rightarrow \overrightarrow{c} = \lambda(2\overrightarrow{a} + 3\overrightarrow{\text{ }d}) }{\Rightarrow \overrightarrow{c} = \lambda(7i - 13j + 19k) }$$Now $(\overrightarrow{a} - \overrightarrow{b}) \cdot \overrightarrow{c} - = \lambda(7 + 52 + 38) + 97\lambda = - 97$
$$\Rightarrow \lambda = - 1 $$Now $\overrightarrow{c} = - 7i + 13j - 19k$
$$\Rightarrow \overrightarrow{c} \times \overrightarrow{k} - = 7j + 13i \Rightarrow |\overrightarrow{c} \times \widehat{k}|^{2} = 7^{2} + 13^{2} = 218$$
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