Arrange the following areas in ascending order

A. Area of parallelogram with adjacent sides $\overline{\mathrm{i}}+2\overline{\mathrm{j}}+3\overline{\mathrm{k}}, 3\overline{\mathrm{i}}-2\overline{\mathrm{j}}+\overline{\mathrm{k}}$ is

B. Area of parallelogram with diagonals $\overline{\mathrm{i}}+2\overline{\mathrm{j}}+3\overline{\mathrm{k}}, 3\overline{\mathrm{i}}-2\overline{\mathrm{j}}+\overline{\mathrm{k}}$ is

C. Area of parallelogram with diagonals are $3\overline{\mathrm{i}}+\overline{\mathrm{j}}-2\overline{\mathrm{k}}, \overline{\mathrm{i}}-3\overline{\mathrm{j}}+4\overline{\mathrm{k}}$ is

D. Area of parallelogram with adjacent sides $3\overline{\mathrm{i}}+\overline{\mathrm{j}}-2\overline{\mathrm{k}}, \overline{\mathrm{i}}-3\overline{\mathrm{j}}+4\overline{\mathrm{k}}$  is

(A)  $\begin{array}{rcl}\overline{\mathrm{a}}& =& \mathrm{i}+2\mathrm{j}+3\mathrm{k} \overline{\mathrm{b}}=3\mathrm{i}-2\mathrm{j}+\mathrm{k}\\ \overline{\mathrm{a}}\times \overline{\mathrm{b}}& =& \left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 2& 3\\ 3& -2& 1\end{array}\right|=\mathrm{i}(2+6)-\mathrm{j}(1-9)+\mathrm{k}(-2-6)\\ & =& 8\mathrm{i}+8\mathrm{j}-8\mathrm{k}\end{array}$

area of $|{|}^{\mathrm{lel}}\mathrm{gram}=|\overline{\mathrm{a}}\times \overline{\mathrm{b}}|=8\sqrt{1+1+1}=8\sqrt{3}$

(B)  area parallel gram $=\frac{1}{2}|\overline{\mathrm{a}}\times \overline{\mathrm{b}}|=4\sqrt{3}$

(C)  $\begin{array}{rcl}\overline{{\mathrm{d}}_1}& =& 3\mathrm{i}+\mathrm{j}-2\mathrm{k} \overline{{\mathrm{d}}_2}=\mathrm{i}-3\mathrm{j}+4\mathrm{k}\\ \overline{{\mathrm{d}}_1}\times \overline{{\mathrm{d}}_2}& =& \left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 3& 1& -2\\ 1& -3& 4\end{array}\right|=\mathrm{i}(-2)-\mathrm{j}\left(14\right)+\mathrm{k}(-10)\\ |\overline{{\mathrm{d}}_1}\times \overline{{\mathrm{d}}_2}|& =& \sqrt{4+196+100}=10\sqrt{3}\end{array}$ 

area of $|{|}^{\mathrm{lel}} \mathrm{gram} =\frac{1}{2}|\overline{{\mathrm{d}}_1}\times \overline{{\mathrm{d}}_2}|=5\sqrt{3}$

(D)  area of $|{|}^{\mathrm{lel}}\mathrm{gram} \overline{{\mathrm{d}}_1}, \overline{{\mathrm{d}}_2}$  adjacent sides

$\begin{array}{rcl} & =& |\overline{{\mathrm{d}}_1}\times \overline{{\mathrm{d}}_2}|=10\sqrt{3}\\ & & \end{array}$