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Question
Find the unit vector in the direction of the sum of the vectors `2hati + 3hatj - hatk and 4hati - 3hatj + 2hatk .`
Solution
\[\text { Let } \vec{a} = 2\hat{ i }+ 3\hat{j} - \hat{k} \text { and }\vec{b} = 4\hat {i}- 3 \hat{j} + 2\hat {k} \]
\[ \vec{c} = \vec{a} + \vec{b} = \left( 2\hat {i} + 3\hat{j} - \hat{k} \right) + \left( 4\hat{i}- 3\hat{j} + 2\hat{k}\right)\]
\[ \Rightarrow \vec{c} = 6 \hat{i}+ \hat{k} \]
\[ \therefore \hat{c} = \frac{\vec{c}}{\left| \vec{c} \right|} = \frac{6 \hat{i}+ \hat{k}}{\sqrt{37}} = \frac{6}{\sqrt{37}} \hat{i}+ \frac{1}{\sqrt{37}} \hat{k}\]
Hence, the required unit vector is \[\frac{6}{\sqrt{37}} \hat{i} + \frac{1}{\sqrt{37}} \hat{k}\] .
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