What are the connotations of "Yisa Hashem Panav Eilecha"?
Rashi: 'Hashem will quash His anger'. 1
Ramban (according to Kabalah): 'Hashem will raise His Face Heavenwards'.
Seforno: 'Hashem will grant you eternal life'. 2
Rashbam: 'Hashem will turn His Face to the one whom He loves to answer his prayers'.
Targum Yonasan: 'Hashem will lift His Face to you when you Daven'. 3
Sifri: It refers to Tefilah - 'If Hashem answered the Tefilah of Lot 4 due to the merit of Avraham, He will certainly answer our Tefilos due to the merit of the Avos. 5
Even when there is good reason to be angry.
Seforno: As the Pasuk writes in Tehilim, 36:10, "Ki Imcha M'kor Chayim, be'Orcha Nir'eh Or", and as the Gemara, describing Olam ha'Ba, states in B'rachos 17a - 'Tzadikim are sitting with their crowns on their heads and deriving pleasure from the glory of the Shechinah'.
See also Ba'al ha'Turim.
Sifri: In Vayeira Bereishis, 19:21, where the Toah wrote "Hinei Nasasi Panecha Gam la'Davar ha'Zeh".
See Torah Twemimah, nmote 152.
What are the connotations of "Veyasem l'cha Shalom"?
Ramban #1: It means 'And He will bless you with peace in your homes ...
Ramban #2 (citing the Sifri): It refers to the peace of Malchus Beis David.
Seforno: It means 'He will give you Eternal peace without interruption and free of punishment'.
Targum Yonasan: It means 'He will grant you peace in all your borders'. 1
Sifri: It means the peace of Torah 2 - as the Pasuk writes in Tehilim, 29:11 - "Hashem Oz le'Amo Yiten Hashem Yevarech es Amo ba'Shalom".
How will we reconcile the current Pasuk "Yisa Hashem Panav Eilecha" with the Pasuk in Eikev Devarim, 10:17 "Asher Lo Yisa Panim"?
Seforno: The Pasuk in Eikev means that Hashem will not favor a sinner to cleanse him from his sins despite the merits of his father.
Moshav Zekeinim #2: Hashem is kindly disposed towards them before the decree is sealed, but not afterwards. 1
B'rachos 20b: In response to the angels' query, Hashem told them that it is correct to favor Yisrael 2 - because, despite the Pasuk "Ve'achalta Vesavata u'Verachta," they recite Birchas ha'Mazon even over a k'Zayis (according to R. Meir) or a k'Beitzah 3 (according to R. Yehudah).
Rosh Hashanah, 17b: The current Pasuk is referring to Mitzvos bein Adam la'Makom whereas the Pasuk in Eikev is referring to Mitzvos Bein Adam la'Chavero. 4
Megilah, 28b: It is due to this question that Birchas Kohanim is read but not translated. 5
Sifri and Nidah, 70b: The current Pasuk is referring to before the decree has been sealed, the Pasuk in Eikev to after the decree has been sealed. 6
Moshav Zekeinim citing R. Shmuel: This would be fine if it said 'Yisa Hashem Panecha.' However, it says "Yisa Hashem Panav Eilecha"! See note on answer #3.
Even though the Torah did not write 'Yisa Hashem Panecha'. See Torah Temimah, note 148.
Divrei Eliyahu and Kol Eliyahu (Devarim 10:17): When a Yisrael has three k'Beitzim, enough for satiation, he prefers to say Birkas ha'Mazon ideally, with a Zimun with Hashem's name (he shares his bread with nine other people, each of whom eats a k'Zayis), or at least a Zimun of three (each one eats a k'Beitzah). Hashem favors Yisrael only when they favor the Torah (when they are meticulous about Mitzvos) ? 'Baruch Hashem who killed him for [that Talmid] did not favor Torah (Shabbos 13b)! Chasam Sofer - when Hashem blesses the food, a very small amount satiates. That suffices to bless ("Vesavata"). However, it says 'Achilah' (which is usually at least a k'Zayis). Due to the concern that the B'rachah will be le'Vatalah, they enacted to eat a full k'Zayis or k'Beitzah before reciting Birchas ha'Mazon.
See Torah Temimah, citing Rosh Hashanah, Ibid., and note 149.
See Torah Temimah on Pasuk 23, citing the Yerushalmi Megilah, note 136.
See Torah Temimah, note 151. Moshav Zekeinim (citing R. Mordechai): The Gemara in Rosh Hashanah, 18a states that Tefilah (of a Tzibur) is effective both before and after the decree is sealed.