Question
Let $\overrightarrow{a} = 2\widehat{i} + \widehat{j} - 2\widehat{k},\overrightarrow{b} = \widehat{i} + \widehat{j}$ and $\overrightarrow{c} = \overrightarrow{a} \times \overrightarrow{b}$. Let $\overrightarrow{d}$ be a vector such that $|\overrightarrow{d} - \overrightarrow{a}| = \sqrt{11},|\overrightarrow{c} \times \overrightarrow{d}| = 3$ and the angle between $\overrightarrow{c}$ and $\overrightarrow{d}$ is $\frac{\pi}{4}$. Then $\overrightarrow{a} \cdot \overrightarrow{d}$ is equal to
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Solution
$\ \overrightarrow{c} = \left| \begin{matrix} \widehat{i} & \widehat{j} & \widehat{k} \\ 2 & 1 & - 2 \\ 1 & 1 & 0 \end{matrix} \right|$
$${\overrightarrow{c} = 2\widehat{i} + 2\widehat{k} + \widehat{k},|c| = 3 }{|\overrightarrow{c} \times \overrightarrow{d}| = 3 }{|\overrightarrow{c}||\overrightarrow{d}|sin\frac{\pi}{4} = 3 \Rightarrow |\overrightarrow{\text{ }d}| = \sqrt{2} }{|\overrightarrow{d} - \overrightarrow{a}| = \sqrt{11} }{\Rightarrow |\overrightarrow{a}|^{2} + |\overrightarrow{d}|^{2} - 2\overrightarrow{a} \cdot \overrightarrow{d} = 11 }{9 + 2 - 2\overrightarrow{a} \cdot \overrightarrow{d} = 11 }{\overrightarrow{a} \cdot \overrightarrow{d} = 0}$$
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