1. **Stating the problem:** We have three scenarios (A, B, C) describing the initial and final positions of two circles on the coordinate plane with arrows indicating their directions. 2. **Understanding the problem:** Initially, the circles are on the x-axis pointing towards each other (one pointing right, the other left). After, they move diagonally either upwards or downwards. 3. **Interpreting the movement:** The arrows change from horizontal to diagonal, indicating a change in velocity direction. We want to describe these movements mathematically. 4. **Representing initial positions:** Let the two circles be at points $(-a,0)$ and $(a,0)$ on the x-axis, with velocities: $$\vec{v}_1 = (v,0), \quad \vec{v}_2 = (-v,0)$$ where $v>0$ is the speed. 5. **Representing final positions:** After movement, the velocities become diagonal: - For scenario A: $\vec{v}_1 = (v, v)$ and $\vec{v}_2 = (-v, -v)$ (up-right and down-left) - For scenario B: $\vec{v}_1 = (v, v)$ and $\vec{v}_2 = (-v, 0)$ (up-right and left) - For scenario C: $\vec{v}_1 = (v, -v)$ and $\vec{v}_2 = (-v, v)$ (down-right and up-left) 6. **Summary:** The problem illustrates changes in velocity vectors from horizontal to diagonal directions. Final answer: The initial velocities are $\vec{v}_1 = (v,0)$ and $\vec{v}_2 = (-v,0)$, and the final velocities for each scenario are: - A: $\vec{v}_1 = (v,v)$, $\vec{v}_2 = (-v,-v)$ - B: $\vec{v}_1 = (v,v)$, $\vec{v}_2 = (-v,0)$ - C: $\vec{v}_1 = (v,-v)$, $\vec{v}_2 = (-v,v)$