https://file.aiquickdraw.com/mathbot/file/1728888958511asbon13g.jpg

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!

Given OA = p and OD = q. Also, OA:AB = 3:2 and 2BC = OD. Find: a) the position vector of C, b) AC, and c) CD.

Learn how to find the position vector of point C and vectors AC and CD in a geometric diagram using vector addition and ratios. The problem involves the vector OA = p, OD = q, and a ratio OA:AB = 3:2. Suitable for students in Grades 10-12.

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