The diagonals of a parallelogram are given by a =2 i - j + k and b = i +3 j - k. Find the area of the parallelogram.

\[\begin{aligned} & \vec{\mathrm{a}} \times \vec{\mathrm{b}}=\left|\begin{array}{ccc} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 2 & -1 & 1 \\ 1 & 3 & -1 \end{array}\right|=-2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+7 \hat{\mathrm{k}} \\ & \text { Area of parallelogram }=\frac{1}{2}|\vec{\mathrm{a}} \times \vec{\mathrm{b}}| \\ & =\frac{1}{2} \sqrt{(-2)^{2}+3^{2}+7^{2}}=\frac{\sqrt{62}}{2} \end{aligned} \]