Question

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

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