1. **Problem Statement:** If the diagonals of a parallelogram are equal, prove that the parallelogram is a rectangle. 2. **Recall the properties and formula:** In a parallelogram, the diagonals bisect each other. Let the parallelogram be $ABCD$ with diagonals $AC$ and $BD$. 3. **Use vector approach:** Let vectors $\vec{AB} = \vec{u}$ and $\vec{AD} = \vec{v}$. Then the diagonals are: $$\vec{AC} = \vec{u} + \vec{v}$$ $$\vec{BD} = \vec{v} - \vec{u}$$ 4. **Given condition:** The diagonals are equal in length, so: $$|\vec{AC}| = |\vec{BD}|$$ Square both sides: $$|\vec{u} + \vec{v}|^2 = |\vec{v} - \vec{u}|^2$$ 5. **Expand both sides:** $$ (\vec{u} + \vec{v}) \cdot (\vec{u} + \vec{v}) = (\vec{v} - \vec{u}) \cdot (\vec{v} - \vec{u}) $$ $$ |\vec{u}|^2 + 2\vec{u} \cdot \vec{v} + |\vec{v}|^2 = |\vec{v}|^2 - 2\vec{u} \cdot \vec{v} + |\vec{u}|^2 $$ 6. **Simplify:** Cancel $|\vec{u}|^2$ and $|\vec{v}|^2$ on both sides: $$ 2\vec{u} \cdot \vec{v} = -2\vec{u} \cdot \vec{v} $$ Add $2\vec{u} \cdot \vec{v}$ to both sides: $$ 4\vec{u} \cdot \vec{v} = 0 $$ So, $$ \vec{u} \cdot \vec{v} = 0 $$ 7. **Interpretation:** The dot product of $\vec{u}$ and $\vec{v}$ is zero, which means $\vec{u}$ is perpendicular to $\vec{v}$. 8. **Conclusion:** Since adjacent sides $AB$ and $AD$ are perpendicular, the parallelogram has a right angle and is therefore a rectangle. **Final answer:** If the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.