Vector geometry and proof

Key learnings

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  Build the vector by following a route

  Any required vector is a sum of journeys: $\vec{AD}=\vec{AO}+\vec{OD}$, where $\vec{AO}=-\vec{OA}$. Express everything in the given base vectors $\mathbf{a}$ and $\mathbf{b}$, then collect terms.

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  Use the section/midpoint formula

  If $M$ divides $AB$ with $AM:MB=m:n$ then $\vec{AM}=\dfrac{m}{m+n}\vec{AB}$. The midpoint is the special case $\tfrac12\vec{AB}$.

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  Parallel ⇔ scalar multiple

  $\vec{CD}=k\,\vec{AB}$ means $CD$ is parallel to $AB$ and $k$ times its length; if they also share a point, the points are collinear. This is how vector "proofs" are argued.