Mathematics - Vector Question with Solution | TestHub

$\overrightarrow{OD}=x\overrightarrow{OA}=x\vec{a}$

If $\vec{a},\vec{b},\vec{c}$ are coplanar, then $l\vec{a}+m\vec{b}+n\vec{c}=0$.

If $\vec{r},\vec{b},\vec{c}$ are collinear, then $\frac{-l}{x}+m+n=0$, so $x=\frac{l}{m+n}$.

$$\overrightarrow{AD}=-x\vec{a}-\vec{a}=-(x+1)\vec{a}$$

Thus, $\frac{OD}{AD}=\frac{x}{x+1}=\frac{l}{l+m+n}$. Similarly, $\frac{OE}{BE}=\frac{m}{l+m+n}$ and $\frac{OF}{CF}=\frac{n}{l+m+n}$.

$$\frac{OD}{AD}+\frac{OE}{BE}+\frac{OC}{CA}=1$$