What is the unit vector perpendicular to the following vectors 2\hat{i} + 2\hat{j} - \hat{k} and 6\hat{i} - 3\hat{j} + 2\hat{k}
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\vec{A} = 2\hat{i} + 2\hat{j} - \hat{k} and \vec{B} = 6\hat{i} - 3\hat{j} + 2\hat{k}
\(\vec{C} = \vec{A} \times \vec{B} = (2\hat{i} + 2\hat{j} - \hat{k}) \times (6\hat{i} - 3\hat{j} + 2\hat{k})\)
\(= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 2 & -1 \\ 6 & -3 & 2 \end{vmatrix} = \hat{i} - 10 \hat{j} - 18 \hat{k}\)
Unit vector perpendicular to both \vec{A} and \vec{B}
\(= \frac{\hat{i} - 10\hat{j} - 18\hat{k}}{\sqrt{1^{2} + 10^{2} + 18^{2}}} = \frac{\hat{i} - 10\hat{j} - 18\hat{k}}{5\sqrt{17}}\)
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