1. **Problem statement:** We need to find the vector $\vec{OX}$ in terms of vectors $\vec{a} = \vec{OA}$ and $\vec{b} = \vec{OB}$, given that point $X$ lies on segment $AB$ such that $AX : XB = 3 : 1$. 2. **Understanding the problem:** Since $X$ lies on $AB$, we can express $\vec{OX}$ as a combination of $\vec{OA}$ and $\vec{OB}$. 3. **Expressing $\vec{OX}$:** Let $\vec{OX} = \vec{OA} + \lambda (\vec{OB} - \vec{OA}) = \vec{a} + \lambda (\vec{b} - \vec{a})$ where $\lambda$ is a scalar between 0 and 1. 4. **Using the ratio $AX : XB = 3 : 1$:** The ratio means $X$ divides $AB$ into 4 equal parts, with $AX$ being 3 parts and $XB$ 1 part. 5. **Finding $\lambda$:** Since $\lambda$ represents the fraction along $AB$ from $A$ to $B$, $\lambda = \frac{AX}{AB} = \frac{3}{3+1} = \frac{3}{4}$. 6. **Substitute $\lambda$ back:** $$\vec{OX} = \vec{a} + \frac{3}{4}(\vec{b} - \vec{a}) = \vec{a} + \frac{3}{4}\vec{b} - \frac{3}{4}\vec{a} = \frac{1}{4}\vec{a} + \frac{3}{4}\vec{b}$$ 7. **Final answer:** $$\boxed{\vec{OX} = \frac{1}{4}\vec{a} + \frac{3}{4}\vec{b}}$$ This means the vector $\vec{OX}$ is a weighted average of $\vec{a}$ and $\vec{b}$ with weights $\frac{1}{4}$ and $\frac{3}{4}$ respectively.