The correct answer is option 3: the vessel more than the plate.

When the metallic plate sticks firmly on the mouth of the water vessel, it is due to the contact between the two metal surfaces. When heat is applied, it causes expansion in the metals.

Option 1: Heat only expanding the vessel is incorrect because both the vessel and the plate will experience expansion.

Option 2: Heat equally expanding both the vessel and the plate is incorrect because different metals have different coefficients of expansion, so they will not expand equally.

Option 3: Heat expanding the vessel more than the plate is the correct answer. Different metals have different coefficients of expansion, and in this case, the vessel has a higher coefficient of expansion compared to the plate. Therefore, when heat is applied, the vessel expands more than the plate, causing the two metal surfaces to separate.

Option 4: Heat expanding the vessel and contracting the plate is incorrect because heat will cause expansion in both the vessel and the plate.