Vectors: Ratio Theorem

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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