Vector Proof Help

[JamieLeaf]

Hey Guys,
I am really stuck on this problem: "OABC is a parallelogram in which P is the midpoint of CB, and D is a point on AP such that d(AD) =2/3d(A/D). Prove that vector OD = 2/3 OB and that O, D and B are collinear."

By the way, does anyone have any advice on how to get better at vector proofs like this and how I could get better at vectors apart from practice? Thanks!!!

[AlphaZero]

Hey Guys,
I am really stuck on this problem: "OABC is a parallelogram in which P is the midpoint of CB, and D is a point on AP such that d(AD) =2/3d(A/D). Prove that vector OD = 2/3 OB and that O, D and B are collinear."

By the way, does anyone have any advice on how to get better at vector proofs like this and how I could get better at vectors apart from practice? Thanks!!!

\(\overrightarrow{OB}=\overrightarrow{OA}+\overrightarrow{OC}\). Now, we just need to find an expression for \(\overrightarrow{OD}\). \begin{align*}\overrightarrow{OD}&=\overrightarrow{OA}+\overrightarrow{AD}\\
&=\overrightarrow{OA}+\frac23\overrightarrow{AP}\\
&=\overrightarrow{OA}+\frac23\left(\overrightarrow{AB}+\overrightarrow{BP}\right)\\
&=\overrightarrow{OA}+\frac23\left(\overrightarrow{AB}-\frac12\overrightarrow{CB}\right)\\
&=\overrightarrow{OA}+\frac23\left(\overrightarrow{OC}-\frac12\overrightarrow{OA}\right)\\
&=\overrightarrow{OA}+\frac23\overrightarrow{OC}-\frac13\overrightarrow{OA}\\
&=\frac23\overrightarrow{OA}+\frac23\overrightarrow{OC}\\
&=\frac23\left(\overrightarrow{OA}+\overrightarrow{OC}\right)\\
&=\frac23\overrightarrow{OB}\end{align*}
Hence, \(O,\ D,\ \text{and}\ B\) are collinear.