A parallelogram is constructed on $3\vec a+\vec b$ and $\vec a - 4\vec b$, where $|\vec a|=6$ and $|\vec b|=8$ and $\vec a$ and $\vec b$ are anti-parallel, then the length of the longer diagonal is:

48

${1_{D_1}}=|4\vec a - 3\vec b|$; $1_{D_2}=|2\vec a +5\vec b|$

$1_{D_1}=16|\vec a|^2+9|\vec b|^2-24\vec . \vec b=\sqrt{16×36+9×64-24×6×8×-1}$

$=\sqrt{576+576+1152}=\sqrt{2304}=48$

$1_{D_2}=\sqrt{4|\vec a|^2+29|\vec b|^2+20\vec . \vec b}=\sqrt{4×36+25×64+20×6×8×-1}=\sqrt{144+1600-960}=\sqrt{784}$

∴ Longer diagonal $(1_{D_1})=48$