To solve the problem and determine the ratio $AP : PB$, let's go through the given information step by step.

Step 1: Define vectors

From the question, we have:

1. $\vec{OA} = 3a$
2. $\vec{OB} = 6b$
3. $\vec{BC} = 5a - b$
4. $\text{The ratio } OP:PC = 2:3$

Let $P$ divide $OC$ in the ratio $2:3$, and express $\vec{OP}$ and $\vec{PC}$ accordingly.

Step 2: Express $\vec{OC}$

From the diagram: $\vec{OC} = \vec{OA} + \vec{AC}$ Since $\vec{AC} = \vec{AB} + \vec{BC}$, we substitute: $\vec{OC} = \vec{OA} + (\vec{AB} + \vec{BC})$ From the question:

• $\vec{OA} = 3a$
• $\vec{AB} = \vec{OB} - \vec{OA} = 6b - 3a$
• $\vec{BC} = 5a - b$

Substitute these values: $\vec{AC} = (6b - 3a) + (5a - b) = 6b - 3a + 5a - b = 4b + 2a$ Thus: $\vec{OC} = \vec{OA} + \vec{AC} = 3a + (4b + 2a) = 5a + 4b$

Step 3: Divide $\vec{OC}$ in the ratio $2:3$

Since $OP:PC = 2:3$, the point $P$ divides $\vec{OC}$ in this ratio. Using the section formula: $\vec{OP} = \frac{3}{5} \vec{OC}, \quad \vec{PC} = \frac{2}{5} \vec{OC}$ Substitute $\vec{OC} = 5a + 4b$: $\vec{OP} = \frac{3}{5}(5a + 4b) = 3a + \frac{12}{5}b$ $\vec{PC} = \frac{2}{5}(5a + 4b) = 2a + \frac{8}{5}b$

Step 4: Express $\vec{AP}$ and $\vec{PB}$

Now calculate $\vec{AP}$ and $\vec{PB}$:

• $\vec{AP} = \vec{OP} - \vec{OA}$
• $\vec{PB} = \vec{OB} - \vec{OP}$

Substitute the known values:

1. $\vec{AP} = (3a + \frac{12}{5}b) - 3a = \frac{12}{5}b$
2. $\vec{PB} = 6b - (3a + \frac{12}{5}b) = -3a + (6b - \frac{12}{5}b) = -3a + \frac{18}{5}b$

Step 5: Find the ratio $AP : PB$

The magnitude of $\vec{AP}$ is: $|\vec{AP}| = \left| \frac{12}{5}b \right| = \frac{12}{5}|b|$

The magnitude of $\vec{PB}$ is: $|\vec{PB}| = \sqrt{(-3a)^2 + \left(\frac{18}{5}b\right)^2} = \sqrt{9a^2 + \frac{324}{25}b^2} = \sqrt{9a^2 + \frac{1296}{25}b^2}$

Thus: $AP : PB = \frac{|\vec{AP}|}{|\vec{PB}|} = \frac{\frac{12}{5}|b|}{\sqrt{9a^2 + \frac{1296}{25}b^2}}$

Simplify further if needed based on $a$ and $b$. Let me know if you'd like assistance continuing from here!