You Do Need power plug adapter while traveling from Faroe Islands to Ecuador.

Here is Why You Need An Adapter

The outlet and plug type used in Ecuador are A and B and in Faroe Islands are C, E, F and K. This is how they look like.

Faroe Islands

In Faroe Islands, outlets of type C, E, F and K are in use.

Type C

Type E

Type F

Type K

Ecuador

In Ecuador, outlets of type A and B are in use.

Type A

Type B