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The two objects were arranged (orthogonal | orthogonally) to each other.

Does "orthogonal" relate to the verb, or to the two objects? In other words, do I need to use the adverb or the adjective?

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    It would take an adverb, but flat adverbs are a thing, so "arranged orthogonal to each other" is not entirely incorrect. (Though some strict grammarians would disagree.) – R.M. May 9 '16 at 20:43
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If you restructure the sentence, you could use either form, but in its current form you should use the adverb because you are describing how the objects are arranged, not simply describing the objects themselves. "Orthogonally" answers the question, "How were the objects arranged?".

The two objects were orthogonal to each other.

The two objects were arranged orthogonally to each other.

The two objects were arranged such that they were orthogonal to each other.

  • "to each other" is redundant here. "Orthogonality" (in the mathematical sense) is by definition a relationship between two objects. If you delete "to each other", it is obvious that "The two objects were arranged orthogonal." is wrong (an adverb should follow the verb). – alephzero May 9 '16 at 18:34
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    In three dimensional space, two objects can be orthogonal to a third, or just towards each other. So the "to each other" is quite an important part, because it distinguishes those cases. – Polygnome May 9 '16 at 19:06
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It's the adverb you need. Orthogonally. Although orthoganality implies arrangement, so you could just say the objects were orthogonal (to each other).

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