Find the vector equation of a line passing through the origin and perpendicular to both the lines \( \vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda(3\hat{i} + 4\hat{j} + 2\hat{k}) \) and \( \vec{r} = \mu(\hat{i} - \hat{j} + \hat{k}) \).
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Whenever a line is specified as being perpendicular to two other lines simultaneously, its direction vector can always be found immediately by computing the cross product of their individual direction vectors.