'arbitrarily-directed' originates from modern English, formed by combining the adverb 'arbitrarily' and the past-participial adjective 'directed'. 'arbitrarily' comes from 'arbitrary' (from Latin via Old French) meaning 'based on one's will or judgment', and 'directed' stems from the verb 'direct' (from Latin 'dirigere') meaning 'to guide or set a direction.'

'arbitrarily' comes from Latin 'arbiter' (via Medieval/Old French 'arbitraire' and Middle English 'arbitrarie'), where shifts gave 'arbitrary' the sense of 'subject to individual will or whim'; 'directed' comes from Latin 'dirigere' through Old French/Latin past-participle forms into Middle English 'direct'. The compound 'arbitrarily-directed' is a modern English formation combining both elements.

Initially the parts 'arbitrary' and 'direct' had senses tied to judgment ('arbiter') and guiding ('dirigere'); over time 'arbitrary' gained the sense 'chosen without fixed rule' and when combined with 'directed' the compound came to mean 'having direction chosen without rule' in contemporary usage.