Let's analyze the problem step by step.

Given:

• $\vec{OA} = \vec{p}$ and $\vec{OD} = \vec{q}$.
• The ratio $OA:AB = 3:2$, meaning $\frac{OA}{AB} = \frac{3}{2}$.
• $2BC = OD$, so $2\vec{BC} = \vec{OD} = \vec{q}$, hence $\vec{BC} = \frac{\vec{q}}{2}$.

a) Find the position vector of C (i.e., $\vec{OC}$):

First, let's express $\vec{OB}$:

• From the ratio $\frac{OA}{AB} = \frac{3}{2}$, we know that $AB = \frac{2}{3} OA$, so: $\vec{AB} = \frac{2}{3} \vec{p}$
• Therefore, the position vector of point B: $\vec{OB} = \vec{OA} + \vec{AB} = \vec{p} + \frac{2}{3} \vec{p} = \frac{5}{3} \vec{p}$

Now, use the fact that $\vec{BC} = \frac{\vec{q}}{2}$ to find $\vec{OC}$:

• $\vec{OC} = \vec{OB} + \vec{BC} = \frac{5}{3} \vec{p} + \frac{\vec{q}}{2}$.

Thus, the position vector of C is: $\vec{OC} = \frac{5}{3} \vec{p} + \frac{1}{2} \vec{q}$

b) Find $\vec{AC}$:

To find $\vec{AC}$, use the relation: $\vec{AC} = \vec{OC} - \vec{OA}$ Substitute the values: $\vec{AC} = \left( \frac{5}{3} \vec{p} + \frac{1}{2} \vec{q} \right) - \vec{p}$ Simplify: $\vec{AC} = \left( \frac{5}{3} \vec{p} - \frac{3}{3} \vec{p} \right) + \frac{1}{2} \vec{q} = \frac{2}{3} \vec{p} + \frac{1}{2} \vec{q}$

c) Find $\vec{CD}$:

To find $\vec{CD}$, use: $\vec{CD} = \vec{OD} - \vec{OC}$ Substitute: $\vec{CD} = \vec{q} - \left( \frac{5}{3} \vec{p} + \frac{1}{2} \vec{q} \right)$ Simplify: $\vec{CD} = -\frac{5}{3} \vec{p} + \left( 1 - \frac{1}{2} \right) \vec{q} = -\frac{5}{3} \vec{p} + \frac{1}{2} \vec{q}$

Would you like any further details or clarifications? Here are five related questions to expand on this:

1. How can the position vectors of other points in the diagram be expressed in terms of $\vec{p}$ and $\vec{q}$?
2. What happens to the position of C if the ratio $OA:AB$ changes?
3. Can you generalize the formula for $\vec{OC}$ if the ratio between $OA$ and $AB$ changes?
4. How would the results change if $2BC \neq OD$ but rather another scalar multiple?
5. Can you derive the length of $AC$ using the position vectors?

Tip: Always start by finding vectors step by step using known points before moving on to other derived vectors!