Vectors $\vec{BC}$ and $\vec{CD}$ as linear combinations of $\vec{AM}$and $\vec{AN}$

Question

How do write the vectors $\vec{BC}$ and $\vec{CD}$ as linear combinations of $\vec{AM}$and $\vec{AN}$, where $M$ and $N$ are the midpoints of the sides $BC$ and $CD$ of a parallelogram $ABCD$.

Answer 1

Hint:

$$ AM ={1\over 2}BC-CD$$ and $$ AN = -{1\over 2}CD+BC$$

Solving this system on $CD$ and $BC$ will give you what you want.

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$$-2AN+AM = -{3\over 2}BC \implies BC = {2\over 3}(2\cdot AN-AM)$$