-(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. -ㄹ까요 sets the thought on the table and lets go; -ㅂ시다 picks it up and walks in front with it. Handing the decision over versus keeping it in your hand.
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?
Setting two coffee cups on the table, taking your hands off them, and looking up at the other person.
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?
Your hand resting on the bag they are carrying, waiting for their eyes to decide whether you lift it.
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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?
Putting tomorrow's weather on the table and nudging it across with a finger.
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?
Laying the phone down on the table and leaving the button for them to press.
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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?
Two color swatches set side by side, and your hands hidden behind your back.
All of it in one picture
-ㄹ까요 is the motion of setting the thought in your head on the table and taking your hands off it. Two coffee cups set down make a suggestion; a hand resting on someone's bag makes an offer to step in; tomorrow's weather nudged across makes a guess you look at together; a phone laid down leaves the permission to them; hands hidden behind two color swatches hands over the choice. In all five the object is on the table, and the last person to touch it is the other one. So asking with -ㄹ까요 does not gather information — it passes the decision across.