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Question
If \[\overrightarrow{a} = \hat{i} + 2 \hat{j} , \vec{b} = \hat{j} + 2 \hat{k} ,\] write a unit vector along the vector \[3 \overrightarrow{a} - 2 \overrightarrow{b} .\]
Solution
Given: \[\overrightarrow{a} = \hat{i} + 2 \hat{j} , \overrightarrow{b} = \hat{j} + 2 \hat{k}\]
Therefore,
\[3 \overrightarrow{a} - 2 \overrightarrow{b} = 3 \hat{i} + 6 \hat{j} - 2 \hat{j} - 4 \hat{k} \]
\[ = 3 \hat{i} + 4 \hat{j} - 4 \hat{k}\]
Hence, Unit vector along \[3 \overrightarrow{a} - 2 \overrightarrow{b} = \frac{3 \hat{i} + 4 \hat{j} - 4 \hat{k}}{\sqrt{3^2 + 4^2 + \left( - 4 \right)^2}} = \frac{3 \hat{i} + 4 \hat{j} - 4 \hat{k}}{\sqrt{9 + 16 + 16}} = \frac{1}{\sqrt{41}} \left( 3 \hat{i} + 4 \hat{j} - 4 \hat{k} \right)\]
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