Q10

Consider three-dimensional vectors $\mathbf a=\overrightarrow{OA}$, $\mathbf b=\overrightarrow{OB}$ and $\mathbf c=\overrightarrow{OC}$ satisfying $\mathbf a\cdot\mathbf b=1$, $\mathbf b\cdot\mathbf c=2$, and $\mathbf c\cdot\mathbf a=3$. Which statement is true?

A. Two of $\mathbf a,\mathbf b,\mathbf c$ could be unit vectors.
B. The points $A,B,C$ could lie on a sphere centred at $O$.
C. For any three-dimensional vector $\mathbf a$, vectors $\mathbf b,\mathbf c$ can be found satisfying the conditions.
D. For all such vectors, there exist positive reals $r,s,t$ such that $r\mathbf a+s\mathbf b+t\mathbf c=\mathbf0$.