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Question
Solution
Given: \[ \vec{a} = 3 \hat{i} - \hat{j} - 4 \hat{k}, \vec{b} = - 2 \hat{i} + 4 \hat{j} - 3 \hat{k}\text{ and } \vec{c} = \hat{i} + 2 \hat{j} - \hat{k}\].
\[\text { Now } , 3 \vec{a} - 2 \vec{b} + 4 \vec{c} = 3\left( 3 \hat{i} - \hat{j} - 4 \hat{k} \right) - 2\left( - 2 \hat{i} + 4 \hat{j} - 3 \hat{k} \right) + 4\left( \hat{i} + 2 \hat{j} - \hat{k} \right)\]
\[ = 9 \hat{i} - 3 \hat{j} - 12\hat{k} + 4 \hat{i} - 8 \hat{j} + 6 \hat{k} + 4 \hat{i} + 8 \hat{j} - 4 \hat{k} \]
\[ = 17 \hat{i} - 3 \hat{j} - 10 \hat{k}\]
Hence,
\[\left| 3 \vec{a} - 2 \vec{b} + 4 \vec{c} \right| = \sqrt{{17}^2 + \left( - 3 \right)^2 + \left( - 10 \right)^2} = \sqrt{289 + 9 + 100} = \sqrt{398}\]
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