$ABCD$ is a parallelogram.
The point $P$ is the midpoint of $AD$. The points $Q$ and $R$ divide $AB$ in three equal parts. ($Q$ is between $A$ and $R$.)
How do I express $PQ$ using only the diagonal vectors $AC$ and $DB$?
I know $ PQ = PA + AQ = \frac{DA}{2} + \frac {AB}{3}$ and $DB = DA + AB = DC + CB$ and $AC = AD + DC = AB + BC$. I have tried expressing my vectors in every possible way but I dont seem to get anywhere. I dont even know what strategy to use to solve these types of problems as I this is my first week of Linear Algebra.
Best Answer
As you wrote, $ \vec PQ = \vec {PA} + \vec {AQ} = \frac{\vec {DA}}{2} + \frac{ \vec {AB}}{3}$
We know diagonals of a parallelogram bisect each other. If $O$ is the intersection point of the diagonals,
$ \displaystyle \small \vec {AB} = \vec {AO} + \vec {OB} = \frac{\vec {AC}}{2} + \frac{\vec {DB}}{2}$
$ \displaystyle \small \vec {DA} = \vec {DO} + \vec {OA} = \frac{\vec {DB}}{2} + \frac{\vec {CA}}{2} = \frac{\vec {DB}}{2} - \frac{\vec {AC}}{2}$
Can you take it from here?