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What is the meaning of the preposition 'in' this below sentence? What relationship does it serve to indicate?

Her house lies in that direction.

Is it idiomatic?

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In is an old word and has many uses.

Cambridge has in preposition (ᴀʀʀᴀɴɢᴇᴍᴇɴᴛ)

used to show how things or people are arranged or divided:

  • We all sat down in a circle.
  • The desks were arranged in rows of ten.

It can be seen that "in that direction" has a similar sense: "We all sat down in that direction"; "The desks were laid out in that direction."

OED has 12b. "Of form, shape, conformation, order", and shows this usage dates from Old English.

So, no: it's not idiomatic. In can be used to show how things are arranged in relationship to each other.

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  • "Her house lies in that direction" is idiomatic. The meaning is "Her house is in that direction." It's about the location of the house, not about the arrangement of multiple objects. Your sentence about desks doesn't demonstrate that the OP sentence is incorrect.
    – gotube
    Aug 31 at 17:56
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Merriam Webster says:

in (preposition)

1a —used as a function word to indicate inclusion, location, or position within limits

  • in the lake
  • wounded in the leg
  • in the summer

In your example, "in the north" indicates location or position within limits

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