1. **State the problem:** Simplify the expression $$\sqrt{2} \cdot \frac{\sqrt{2}}{3} + \sqrt{3} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3}$$. 2. **Recall the rule:** The product of square roots is the square root of the product: $$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$. 3. **Simplify each term:** - First term: $$\sqrt{2} \cdot \frac{\sqrt{2}}{3} = \frac{\sqrt{2} \cdot \sqrt{2}}{3} = \frac{\sqrt{4}}{3} = \frac{2}{3}$$. - Second term: $$\sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3} \cdot \sqrt{3}}{2} = \frac{\sqrt{9}}{2} = \frac{3}{2}$$. - Third term: $$\frac{\sqrt{3}}{3} \cdot \frac{\sqrt{3}}{3} = \frac{\sqrt{3} \cdot \sqrt{3}}{3 \cdot 3} = \frac{\sqrt{9}}{9} = \frac{3}{9} = \frac{1}{3}$$. 4. **Combine all terms:** $$\frac{2}{3} + \frac{3}{2} - \frac{1}{3}$$. 5. **Find common denominator:** The common denominator of 3 and 2 is 6. 6. **Rewrite each fraction:** $$\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}$$ $$\frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6}$$ $$\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$$ 7. **Perform addition and subtraction:** $$\frac{4}{6} + \frac{9}{6} - \frac{2}{6} = \frac{4 + 9 - 2}{6} = \frac{11}{6}$$. **Final answer:** $$\boxed{\frac{11}{6}}$$.