A parallelogram is constructed on  $3\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}}-4\overrightarrow{\mathrm{b}}$, where $\left|\overrightarrow{\mathrm{a}}\right|=6$ and  $\left|\overrightarrow{\mathrm{b}}\right|=8$ and $\overrightarrow{\mathrm{a}}\mathrm{and}\overrightarrow{\mathrm{b}}$ are anti parallel. Then the length of the longer diagonal is

From the diagram 
$\begin{array}{l}\overline{\mathrm{AC}}=(3\overline{\mathrm{a}}+\overline{\mathrm{b}})+(\overline{\mathrm{a}}-4\overline{\mathrm{b}})=4\overline{\mathrm{a}}-3\overline{\mathrm{b}}\\ \left|\overline{\mathrm{AC}}\right|=\sqrt{16|\overline{\mathrm{a}}{|}^2+9|\overline{\mathrm{b}}{|}^2-24\overline{\mathrm{a}.}\overline{\mathrm{b}}}\\ =\sqrt{16\left(36\right)+9\left(64\right)+24\left(6\right)\left(8\right)}\\ =\sqrt{{\left(48\right)}^2} \\ =48(\because \overline{\mathrm{a}}\text{ and }\overline{\mathrm{b}}\text{ are anti parallel, }\therefore \overline{\mathrm{a}.}\overline{\mathrm{b}}=-|\overline{\mathrm{a}}|\overline{\mathrm{b}}\mid )\\ and \overline{BD}=\overline{BA}+\overline{AD}=-(3\overline{\mathrm{a}}+\overline{\mathrm{b}})+(\overline{\mathrm{a}}-4\overline{\mathrm{b}})\\ \overline{BD}=-(2\overline{a}+5\overline{b})\\ \left|\overline{BD}\right|=\sqrt{4{\left|\overline{a}\right|}^2+25{\left|\overline{b}\right|}^2+20(\overline{a}.\overline{b})}\\ =\sqrt{4\left(36\right)+25\left(64\right)-20\left(6\right)\left(8\right)}\\ =\sqrt{784}\\ =\sqrt{{\left(28\right)}^2} \\ =28\\ \therefore length of longer diagonal=48\end{array}$