When I took this (end of year) test, back in June, I found that there were many topics we hadn't covered yet, so I took it upon my self to teach myself over the summer each the topics enough to be able to answer their corresponding question(s), which I have achieved on all but one question. This one.
At first, I simply tried approaching the question as a regular triangle question. I then proceeded to have many failed attempts with trigonometry, and Pythagoras, and Heron's Formula. Next I learnt about vectors, and tried applying that knowledge, and continued to fail.
I'm sure this will be obvious to some, but I just cannot figure it out.
The following is the question exactly as is it on the test.
https://i.stack.imgur.com/khuAn.png
In the diagram,
$\vec{OB}^{\,} = b$
$\vec{OC}^{\,} = c$
$\vec{OC}^{\,} = \frac{1}{3}\vec{OA}^{\,}$
$\vec{BD}^{\,} = \frac{1}{4}\vec{BA}^{\,}$
Find $CD$ in terms of b and c.
Give your answer in its simplest form.
You must show all your working.
P.S. Apologies for the unembeded image, I'm new here.
Hint: $\;\overrightarrow{CD} = \overrightarrow{CO} + \overrightarrow{OB} + \overrightarrow{BD} = -c + b + \frac{1}{4} \overrightarrow{BA}\,$, and $\,\overrightarrow{BA} = \overrightarrow{OA} - \overrightarrow{OB} = 3c - b\,$.