Matot - The Vengeance of the Lord

A quick thought or two:
Moshe is told to take Israel's revenge on Midian. Bemidbar 31:2-3:

ב נְקֹם, נִקְמַת בְּנֵי יִשְׂרָאֵל, מֵאֵת, הַמִּדְיָנִים; אַחַר, תֵּאָסֵף אֶל-עַמֶּיךָ.	2 'Avenge the children of Israel of the Midianites; afterward shalt thou be gathered unto thy people.'
ג וַיְדַבֵּר מֹשֶׁה אֶל-הָעָם לֵאמֹר, הֵחָלְצוּ מֵאִתְּכֶם אֲנָשִׁים לַצָּבָא; וְיִהְיוּ, עַל-מִדְיָן, לָתֵת נִקְמַת-ה, בְּמִדְיָן.	3 And Moses spoke unto the people, saying: 'Arm ye men from among you for the war, that they may go against Midian, to execute the LORD'S vengeance on Midian.

And, as a midrash Tanchuma (and Rashi) points out, despite the statement that afterwards Moshe's life would be over, he still immediately goes out to do it. This is contrasted to Yehoshua who in a similar situation delays. Without this midrash could also view it as the fact that Moshe's time is coming up, regardless, and Hashem is letting him finish up with this final act, to get closure.

The transition from Hashem's perspective and the human perspective is also of interest. Hashem says נִקְמַת בְּנֵי יִשְׂרָאֵל while Moshe says לָתֵת נִקְמַת-ה. Each, we might say, is motivated by concern for the other -- Hashem angered by what they did to the Israelites, and Moshe/Israel because of bringing them to sin was an attack on Hashem via his people. Rashi/Midrash Tanchuma explains the shift:

For anyone opposing Israel is reckoned as opposing the Holy One, blessed is He.

Alternatively, we might say that נִקְמַת-ה is not vengeance for an offense done to Hashem, but rather this is taking action on behalf of Hashem, for whatever cause Hashem has to bring vengeance -- in this case, because of the hurt done to Israel. Thus, the army is God's proxy for taking revenge on behalf of Israel, and thus the shift.

The phrase nikmat Hashem occurs elsewhere, in Yirmeyahu 50 and 51. In Yirmeyahu 50:

יד עִרְכוּ עַל-בָּבֶל סָבִיב, כָּל-דֹּרְכֵי קֶשֶׁת, יְדוּ אֵלֶיהָ, אַל-תַּחְמְלוּ אֶל-חֵץ: כִּי לַה, חָטָאָה.	14 Set yourselves in array against Babylon round about, all ye that bend the bow, shoot at her, spare no arrows; for she hath sinned against the LORD.
טו הָרִיעוּ עָלֶיהָ סָבִיב, נָתְנָה יָדָהּ, נָפְלוּ אשויתיה (אָשְׁיוֹתֶיהָ), נֶהֶרְסוּ חוֹמוֹתֶיהָ: כִּי נִקְמַת ה הִיא הִנָּקְמוּ בָהּ, כַּאֲשֶׁר עָשְׂתָה עֲשׂוּ-לָהּ.	15 Shout against her round about, she hath submitted herself; her buttresses are fallen, her walls are thrown down; for it is the vengeance of the LORD, take vengeance upon her; as she hath done, do unto her.

and this is because of what they have done to Israel:

י וְהָיְתָה כַשְׂדִּים, לְשָׁלָל: כָּל-שֹׁלְלֶיהָ יִשְׂבָּעוּ, נְאֻם-ה.	10 And Chaldea shall be a spoil; all that spoil her shall be satisfied, saith the LORD.
יא כִּי תשמחי (תִשְׂמְחוּ) כִּי תעלזי (תַעַלְזוּ), שֹׁסֵי נַחֲלָתִי: כִּי תפושי (תָפוּשׁוּ) כְּעֶגְלָה דָשָׁה, ותצהלי (וְתִצְהֲלוּ) כָּאַבִּרִים.	11 Because ye are glad, because ye rejoice, O ye that plunder My heritage, because ye gambol as a heifer at grass, and neigh as strong horses;

So it is Hashem's vengeance on behalf of klal Yisrael.

In the next perek, Yirmeya 51, we have:

יא הָבֵרוּ הַחִצִּים, מִלְאוּ הַשְּׁלָטִים--הֵעִיר יְהוָה אֶת-רוּחַ מַלְכֵי מָדַי, כִּי-עַל-בָּבֶל מְזִמָּתוֹ לְהַשְׁחִיתָהּ: כִּי-נִקְמַת יְהוָה הִיא, נִקְמַת הֵיכָלוֹ.

11 Make bright the arrows, fill the quivers, the LORD hath roused the spirit of the kings of the Medes; because His device is against Babylon, to destroy it; for it is the vengeance of the LORD, the vengeance of His temple.

The Medes are not doing this because of some insult to Hashem. Rather, this is Hashem's way of taking vengeance, through a human army, because of some injury done -- to His temple, and perhaps to Israel in the process.

(though we could cast this as a way of Hashem's getting vengeance for a wrong done to Hashem, citing Rashi/Tanchuma.)