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Question
The adjacent sides of a parallelogram are represented by the vectors \[\vec{a} = \hat{i} + \hat{j} - \hat{k}\text{ and }\vec{b} = - 2 \hat{i} + \hat{j} + 2 \hat{k} .\]
Find unit vectors parallel to the diagonals of the parallelogram.
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Solution
\[\vec{a} = \hat{i} + \hat{j} - \hat{k} \text{ and }\vec{b} = - 2 \hat{i} + \hat{j} + 2 \hat{k} .\]
\[\overrightarrow{AC} = \vec{a} + \vec{b} = - \hat{i} + 2 \hat{j} + \hat{k}\]
\[\text{Let the unit vector along the diagonals AC and BD of the parallelogram be }\widehat{AC}\text{ and }\widehat{BD} . \]
\[ \Rightarrow \widehat{AC} = \frac{- \hat{i} + 2 \hat{j} + \hat{k}}{\sqrt{6}}\]
\[\Rightarrow \widehat{BD} = \frac{- 3 \hat{i} + 3 \hat{k}}{3\sqrt{2}} = \frac{- \hat{i} + k}{\sqrt{2}}\]
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