Question
Let $\overrightarrow{a} = - \widehat{i} + \widehat{j} + 2\widehat{k},\overrightarrow{b} = \widehat{i} - \widehat{j} - 3\widehat{k},\overrightarrow{c} = \overrightarrow{a} \times \overrightarrow{b}$ and $\overrightarrow{d} = \overrightarrow{c} \times \overrightarrow{a}$. Then $(\overrightarrow{a} - \overrightarrow{b}) \cdot \overrightarrow{d}$ is equal to :
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Solution
$\overrightarrow{d} = (\overrightarrow{a} \times \overrightarrow{b}) \times \overrightarrow{a}$
$${\overrightarrow{d} = \left( a^{2} \right)\overrightarrow{b} - (\overrightarrow{a} \cdot \overrightarrow{b})\overrightarrow{a} }{\overrightarrow{d} = 6\overrightarrow{\text{ }b} + 8\overrightarrow{a} }{(\overrightarrow{a} - \overrightarrow{b}) \cdot \overrightarrow{d} - = (\overrightarrow{a} - \overrightarrow{b}) \cdot (6\overrightarrow{b} + 8\overrightarrow{a}) }{= 8a^{2} - 6{\text{ }b}^{2} - 2\overrightarrow{a} \cdot \overrightarrow{b} }{= 48 - 66 + 16 = - 2}$$
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