1. **State the problem:** Find the magnitude (length) of the vector or segment $\overrightarrow{AB}$ using the Pythagorean theorem. 2. **Formula:** The Pythagorean theorem states that for a right triangle with legs of lengths $a$ and $b$, and hypotenuse $c$, we have: $$c = \sqrt{a^2 + b^2}$$ 3. **Explanation:** If $A$ and $B$ are points in a coordinate plane with coordinates $A(x_1, y_1)$ and $B(x_2, y_2)$, then the length of $\overrightarrow{AB}$ is the distance between these points: $$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 4. **Intermediate work:** Suppose the horizontal distance is $\Delta x = x_2 - x_1$ and vertical distance is $\Delta y = y_2 - y_1$. 5. **Apply the formula:** $$AB = \sqrt{(\Delta x)^2 + (\Delta y)^2}$$ 6. **Simplify:** Calculate the squares, add them, then take the square root. 7. **Final answer:** The magnitude of $\overrightarrow{AB}$ is $$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ This formula gives the length of the segment $AB$ using the Pythagorean theorem.