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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