1. The farmer uses $\frac{1}{3}$ of his land to plant cassava.\n\n2. The remaining land after planting cassava is $1 - \frac{1}{3} = \frac{2}{3}$.\n\n3. The farmer uses $\frac{1}{3}$ of this remaining $\frac{2}{3}$ to plant maize, which is $\frac{1}{3} \times \frac{2}{3} = \frac{2}{9}$.\n\n4. So the total land used for cassava and maize together is $\frac{1}{3} + \frac{2}{9} = \frac{3}{9} + \frac{2}{9} = \frac{5}{9}$.\n\n5. The remaining land used for vegetables is $1 - \frac{5}{9} = \frac{4}{9}$.\n\nFinal answer: The farmer used $\frac{4}{9}$ of his land to plant vegetables.