도무지 and 도저히 are synonyms used in negative expressions, often but not always interchangeable. * 도무지: for all I do, for all I know; even if I try everything I can think of. * 도저히: though I'm giving all I got; no matter how hard I try; for the life of me. 도무지 emphasizes the various attempts made, something not going one's way in spite of all the tries. 도저히 is more about the intensity and magnitude of the effort. 도무지 implies a sense of bewilderment or being at a loss, while 도저히 suggests having no results despite great efforts for a clearly defined goal. So 도무지 is often used in connection with lighter things like knowledge and memory, while 도저히 tend to involve more serious endeavors. Examples: - 열쇠를 어디다 두었는지 도무지 기억이 안 난다: I can't seem to remember where I put the keys. - 이 창문이 도무지 안 열린다: This window just won't open for some reason. - 그 사람은 도무지 예의라곤 없다: He altogether lacks the concept of an etiquette. (not interchangeable) - 여러 번 시도했는데 도저히 안 되네요: I tried so many times but just can't make it work. - 이런 대우는 도저히 참고 넘길 수 없다: I really can't stand this kind of treatment anymore. (not interchangeable)

전혀, 도무지 - Not at all 도저히 - Completely, Absolutely, Totally They have almost same meaning which is a nagative meaning. But 전혀 and 도무지 have a more nagative meaning. More or less, they have very similar meaning. I hope my explanation is helpful to you.

Hi Lindsay, I referenced a monolingual Korean dictionary native speakers would use and it gives very similar definitions for both words. 도무지 [부사] 1. 아무리 해도.; 2. 이러니저러니 할 것 없이 아주. 도저희 [부사] 아무리 하여도. I think both words can be used interchangeably in pretty much all situations. Can't think of one where only one could be used (and the other couldn't be used). Hope that helps.