What exactly, is the "Yad ha'Gedolah"?
Rashi: It is synonymous with the great might with which Hashem dealt with the Egyptians.
Ramban #1 (citing Targum Onkelos): It refers to the Might of Hashem's Right Hand (the Midas ha'Din 1 ), with which He struck the Egyptians.
Ramban #2 (citing the Ibn Ezra): Yad ha'Gedolah is another way of saying the great plague (stroke). 2
Ramban #3 (according to Kabalah): It is written with reference to Hashem's Midas ha'Din. 3
Targum Yonasan: It refers to the Mighty Hand of Hashem with which He performed miracles in Egypt.
Whilst it was with the Midas Chesed of the same Hand that He saved Yisrael (See Rav Chavel's notes).
Similar to the Pasuk in Va'eira (9:3) "Hinei Yad Hashem Hoyah be'Mikn'cha" (Ramban).
What are the ramifications of the fact that they believed in Hashem?
Rashbam: They now knew that they would not die of starvation in the desert.