To find the vector $\overrightarrow{OR}$, we need to use the given information from the diagram and the problem:

1. Understanding the relationships:

   • The triangle $OAB$ is described using vectors: $\overrightarrow{OA} = 10\mathbf{a}, \quad \overrightarrow{OB} = 10\mathbf{b}.$
   • $ARQ$ and $ORP$ are straight lines.
   • The vectors are simplified in terms of fractions along the lines, e.g., $\overrightarrow{OQ} = \frac{1}{5}\overrightarrow{OB} = 2\mathbf{b}$.

2. Position of $R$ along $\overrightarrow{OR}$:

   • From the diagram and the problem's symmetry, the vector $\overrightarrow{R}$ splits $\overrightarrow{OP}$ in some ratio, so we need to establish $R$ along the straight line.

3. Calculating $\overrightarrow{OR}$: If $\overrightarrow{R} = k \overrightarrow{OA} + (1 - k)\overrightarrow{OQ}$, calculate from known ratios. Let me work it out. Vector definitions and relationships import sympy as sp

Define vectors

a, b, k = sp.symbols('a b k') # a, b are vector directions, k is scalar ratio

OA and OQ in terms of given data

OA = 10 * a OB = 10 * b OQ = 2 * b # from the problem statement

OR in terms of OA and OQ, let k represent the fraction along the line

OR = k * OA + (1 - k) * OQ

Simplify OR

OR_simplified = sp.simplify(OR) OR_simplified