Mathematics - Vector Question with Solution | TestHub

Let the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$. Then $\overrightarrow{O{G}_1}=(\vec{a}+\vec{b}+\vec{c})/3$ and $\overrightarrow{B{G}_2}=(\vec{a}+\vec{c}-3\vec{b})/3$. Their dot product is zero. Using $|\vec{c}-\vec{a}{|}^2=b^2,|\vec{c}-\vec{b}{|}^2=a^2$, and $|\vec{a}-\vec{b}{|}^2=c^2$, simplification gives $2a^2+2c^2-6b^2=0$. Hence the ratio is 3 .