The image you've uploaded contains a geometry problem involving triangle OAB. It includes vector notations and several key points such as M, N, and P. Here's a breakdown of the given information:

• AB vector: $\overrightarrow{AB} = -8\mathbf{a} + 6\mathbf{b}$
• BN vector: $\overrightarrow{BN} = -\frac{8}{3}\mathbf{a} + \frac{6}{3}\mathbf{b}$
• The segment OM:MB is in a ratio of 1:2.
• N is the midpoint of AB.
• The point P is the intersection of lines ON and AM.
• The goal is to find the vector $\overrightarrow{OP}$ as a simplified expression in terms of $\mathbf{a}$ and $\mathbf{b}$.

To solve this, we will use the vector method for division of a line segment, and the intersection of two lines.

Step 1: Find the position of point M

Since M divides OB in the ratio $OM:MB = 1:2$, we can express the position of point M as: $\overrightarrow{OM} = \frac{1}{3} \overrightarrow{OB}$ Thus, $\overrightarrow{OM} = \frac{1}{3} (8\mathbf{a} + 6\mathbf{b})$ $\overrightarrow{OM} = \frac{8}{3}\mathbf{a} + 2\mathbf{b}$

Step 2: Find the position of point N

Since N is the midpoint of AB, we have: $\overrightarrow{ON} = \frac{1}{2} \overrightarrow{AB}$ Thus: $\overrightarrow{ON} = \frac{1}{2} (-8\mathbf{a} + 6\mathbf{b})$ $\overrightarrow{ON} = -4\mathbf{a} + 3\mathbf{b}$

Step 3: Find the equation of line ON

To express line ON in parametric form, we assume a parameter $t$ such that any point on ON is: $\overrightarrow{OP} = \overrightarrow{O} + t (\overrightarrow{ON} - \overrightarrow{O}) = t(-4\mathbf{a} + 3\mathbf{b})$ Thus, the position vector of a point on ON is: $\overrightarrow{OP} = t (-4\mathbf{a} + 3\mathbf{b})$

Step 4: Find the equation of line AM

To express line AM, we assume a parameter $s$ such that any point on AM is: $\overrightarrow{AP} = \overrightarrow{A} + s (\overrightarrow{AM} - \overrightarrow{A}) = s (\overrightarrow{AM} )$ Thus, the position vector of a point on AM is given by simplifying $P$