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Task: In parallelogram OABC sides AB and BC have the centers D and E. Prove with vectors that vectors OD and OE divide the Diagonal vector AC into three equal parts.

I got this task from my teacher and i couldn't do it and i'd just like to see how it is supposed to be done just for my interest.

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Have you seen any other problems in which you have to find out where two vectors intersect?

Rather than showing you the solution, here are some steps to try.

In two dimensions you need two basis vectors - I'd suggest using OA and OB and writing everything in terms of those.

Then name the point at which OD meets AC - say F.

Then put OF = xOD and AF = yAC. Written in terms of basis vectors you get two equations in the two unknowns x and y, which you should be able to solve.

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