Final answer:
The atomic radius increases down a group and decreases across a period; thus, the order of increasing radius is vanadium, zinc, potassium, and bromide.
Step-by-step explanation:
To arrange the given elements and ions—bromide (Brⁱ), potassium (K), vanadium (V), and zinc (Zn)—in order of increasing atomic radius, we consider their positions in the periodic table. The atomic radius increases as we move down a group and decreases as we move across a period from left to right. Potassium is in group 1 and period 4, vanadium is in group 5 and period 4, and zinc is in group 12 and period 4. As for the bromide ion, it is a singly charged negative ion, which typically has a larger radius than its neutral atom due to the additional electron and increased electron-electron repulsion.
Following these trends, the order from smallest to largest radius is: vanadium < zinc < potassium < bromide. Vanadium and zinc are closer to the right of the period, thus smaller than potassium. The bromide ion, being an anion, will have a larger radius than the potassium atom.