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Question
If \[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k} \text{ and }\vec{c} = \hat{i} - 2 \hat{j} + \hat{k} ,\] find a unit vector parallel to \[2 \vec{a} - \vec{b} + 3 \vec{c .}\]
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Solution
We have, \[\vec{a} = \hat{i} + \hat{j} + \hat{k}\] \[\vec{b} = 2 \hat{i} - \hat{j} + 3 \hat{k}\] and \[\vec{c} = \hat{i} - 2 \hat{j} + \hat{k}\]
∴ \[2 \vec{a} - \vec{b} + 3 \vec{c} = 2\left( \hat{i} + \hat{j} + \hat{k} \right) - \left( 2 \hat{i} - \hat{j} + 3 \hat{k} \right) + 3\left( \hat{i} - 2 \hat{j} + \hat{k} \right) = 3 \hat{i} - 3 \hat{j} + 2 \hat{k} .\]
A unit vector parallel to \[2 \vec{a} - \vec{b} + 3 \vec{c}\] is given by \[\frac{2 \vec{a} - \vec{b} + 3 \vec{c}}{\left| 2 \vec{a} - \vec{b} + 3 \vec{c} \right|}\]
\[= \frac{\left( 3 \hat{i} - 3 \hat{j} + 2 \hat{k} \right)}{\sqrt{22}}\]
\[ = \frac{3}{\sqrt{22}} \hat{i} - \frac{3}{\sqrt{22}} \hat{j} + \frac{2}{\sqrt{22}} \hat{k}\]
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