Question

\[\text{ If } \overrightarrow{a} = 3 \hat{i} - \hat{j} - 4 \hat{k} , \overrightarrow{b} = - 2 \hat{i} + 4 \hat{j} - 3 \hat{k} \text{ and }\overrightarrow{c} = \hat{i} + 2 \hat{j} - \hat{k} ,\text{ find }\left| 3 \overrightarrow{a} - 2 \overrightarrow{b} + 4 \overrightarrow{c} \right| .\]

Answer 1

Given  \[\overrightarrow{a} = 3 \hat{i} - \hat{j} - 4 \hat{k} , \overrightarrow{b} = - 2 \hat{i} + 4 \hat{j} - 3 \hat{k}, \overrightarrow{c} = \hat{i} + 2 \hat{j} - \hat{k}\]
Now,
\[3 \overrightarrow{a} - 2 \overrightarrow{b} + 4 \overrightarrow{c} = 3\left( 3 \hat{i} - \hat{j} - 4 \hat{k} \right) - 2\left( - 2 \hat{i} + 4 \hat{j} - 3 \hat{k} \right) + 4\left( \hat{i} + 2 \hat{j} - \hat{k} \right)\]
\[9 \hat{i} - 3 \hat{j} - 12 \hat{k} + 4 \hat{i} - 8 \hat{j} + 6 \hat{k} + 4 \hat{i} + 8 \hat{j} - 4 \hat{k}\]
\[17 \hat{i} - 3 \hat{j} - 10 \hat{k}\]
∴\[\left| 3 \overrightarrow{a} - 2 \overrightarrow{b} + 4 \overrightarrow{c} \right| = \sqrt{{17}^2 + \left( - 3 \right)^2 + \left( - 10 \right)^2} = \sqrt{289 + 9 + 100} = \sqrt{398}\]