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Let \(\vec{a}=\hat{i}+4\hat{j}+2\hat{k},\) \(\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}\) and \(\vec{c}=2\hat{i}-\hat{j}+4\hat{k}.\)

Find a vector \(\vec{p}\) which is perpendicular to both \(\vec{a}\) and \(\vec{b}\) and \(\vec{p}.\vec{c}=18.\)

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Given, \(\vec{a}=\hat{i}+4\hat{j}+2\hat{k},\) \(\vec{b}=3\hat{i}-2\hat{j}+7\hat{k},\) \(\vec{c}=2\hat{i}-\hat{j}+4\hat{k}\)

Vector \(\vec{p}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\) \(i.e,\) \(\vec{p}\) is parallel to vector \(\vec{a}\,\times ​​\vec{b}.\)

Since \(\vec{p}\) is parallel to \(\vec{a}\,\times ​​\vec{b}\)

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