-(으)ㄹ까요
-(eu)lkkayo
Setting your thought down in front of the other person
The picture Taking the thought out of your own head, placing it on the table in front of someone, and pushing the decision over to their side.
Where the picture shows up in -(으)ㄹ까요
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putting a suggestion on the table
우리 커피 마실까요?
u-ri keo-pi ma-sil-kka-yo
Shall we go for coffee?
마셔요 has already decided; 마실까요 leaves the choice with them.
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asking whether you should step in
제가 도와드릴까요?
je-ga do-wa-deu-ril-kka-yo
Shall I give you a hand?
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putting a guess on the table
내일 비가 올까요?
nae-il bi-ga ol-kka-yo
Do you think it'll rain tomorrow?
It doesn't matter that nobody knows — you're inviting them to think it through with you.
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leaving permission to them
지금 전화해도 될까요?
ji-geum jeon-hwa-hae-do doel-kka-yo
Would it be okay if I called now?
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handing the choice over
이 색이 나을까요?
i sae-gi na-eul-kka-yo
Do you think this color is better?