What is "Makas Cherev v'Hereg v'Avdan"?
Vilna Gaon: Initially they struck them via the sword in a place that will not kill them, in order to increase their affliction 1 . Afterwards they killed them, and afterwards they disposed of the bodies, lest the Goyim always remember.
Taima d'Kra (Rav C. Kanievsky): Cruelty is not Yisrael's Midah! Amatzyah was punished for casting Bnei Edom from a rock (Eichah Rabah 14)! However, we find "va'Yshasef Shmuel Es Agag" (Shmuel I, 15:33) - he cut him into small pieces, and cast them to ostriches (Eichah Rabah 3). Hashem command Yehoshua to fight Amalek without disgracing them (Mechilta). If not, he would have disgraced them! Amalek is unlike others; they primarily wanted to fight Hashem. They cut off Milos of Yisrael and cast them towards Shamayim. It is a Mitzvah to do to them like their intent, Midah k'Neged Midah (refer to 9:5:2:1).
What is "ki'Rtzonam"?
Vilna Gaon: They did to their haters what their haters wanted to do to them 1 ; they killed them.
Malbim: They belittled them and lowered them. They were permitted only to take vengeance on "Oyveihem" (8:13), i.e. those who openly hate them, but not against Son'eihem, whose anger is hidden.
