The area of the parallelogram whose adjacent sides are given by the vectors \(\vec{a}=3\hat{i}+\hat{j}+4\hat{k}\) and \(\vec{b}=\hat{i}-\hat{j}+\hat{k}\) is

\(\sqrt{21}\) \(\frac{1}{2}\sqrt{21}\) \(\sqrt{42}\) \(\frac{\sqrt{42}}{2}\)

\(\sqrt{42}\)

\(\begin{aligned}\text{Area}&=|\vec{a}\times \vec{b}|\\ \vec{a}\times \vec{b}&=\left|\begin{array}{lll}\hat{i}& \hat{j}& \hat{k}\\ 3& 1&4\\ 1&-1 & 1\end{array}\right|\\ &=5\hat{i}+\hat{j}+\hat{k}(-4)\\ \text{Area}&=|\vec{a}\times \vec{b}|\\ &=\sqrt{25+1+16}\\ &=\sqrt{42}\end{aligned}\)