Question:

Which must be true: I) $\vec{a}=2\hat{i}+\hat{j}+\hat{k}$, $\vec{b}$ and $\vec{c}$ nonzero such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b}\cdot\vec{c}=0$, then $|\vec{a}+\lambda\vec{c}|\ge|\vec{a}|$ for all $\lambda\in\mathbb{R}$. II) If $\overrightarrow{PQ},\overrightarrow{QR},\overrightarrow{RS},\overrightarrow{ST},\overrightarrow{TU}$ and $\overrightarrow{UP}$ represent sides of regular hexagon, then $\overrightarrow{PQ}\times(\overrightarrow{RS}+\overrightarrow{ST})\ne\vec{0}$. III) Four points $A,B,C,D$ with position vectors $\vec{a},\vec{b},\vec{c},\vec{d}$ are coplanar, then there exist constants $x,y,z,w$ such that $x\vec{a}+y\vec{b}+z\vec{c}+w\vec{d}=\vec{0}$ with $x+y+z+w=0$ but not all zero.