As we all know, the Ratio Theorem is given in MF15. But some students struggle to fit in the $\lambda$ and $\mu$ every time. They will then attempt to draw a bit and see it helps. Here’s a solution for students who can never figure out the idea of ratio.

Given $AB:BC = 2:3$, we have that $\frac {AB}{BC} = \frac {2}{3}$. Notice that you canNOT take a ratio of vectors, so we don’t write it as $\vec{AB}$ or $\vec{BC}$! If we simply cross multiply the above, we have $3\vec{AB}=2\vec{BC}$. From here,

$3(\vec{OB}-\vec{OA})=2(\vec{OC}-\vec{OB})$

$3\vec{OB}-3\vec{OA}=2\vec{OC}-2\vec{OB}$

And depending on what vector we are interested, its just a simple manipulation and substitution from here. You will also notice that you can “bump” into the ratio theorem here.

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