Animal heading off in a particular direction (4)
I believe the answer is:
east
'in a particular direction' is the definition.
'animal heading off' is the wordplay.
'animal' becomes 'beast' (synonyms).
'heading off' means to remove the first letter.
'beast' with its first letter taken away is 'EAST'.
(Other definitions for east that I've seen before include "To right if facing North" , "Oriental lands" , "Where the sun rises" , "English region" , "Bearing of 90 degrees" .)