What was the Plague of the Sword?

Question

In the Haggadah you will find the following exegesis of part of Deuteronomy 26:8:

וּבִזְרֹעַ נְטוּיָה. זוֹ הַחֶרֶב, כְּמָה שֶּׁנֶּאֱמַר: וְחַרְבּוֹ שְׁלוּפָה בְּיָדוֹ, נְטוּיָה עַל-יְרוּשָלָיִם..‏
"And with an outstretched forearm" - this [refers to] the sword, as it is stated (I Chronicles 21:16); "And his sword was drawn in his hand, leaning over Jerusalem:

• What is the sword that is referred to? The only references to swords I can see in Exodus are 5:3, 5:21 and 18:4 and all of those are threats of sword-use against the Jews. How did God use a sword with his outstretched arm to help redeem the Jewish slaves (recall we are expounding the phrase "[God took us out] with an outstretched arm")?

• Why are we referencing such a seemingly unrelated part of Tanakh? The story of the plague following King David's census seems completely unrelated to the Exodus. Is there a thematic connection?

I note the phrase "חרבו שלופה בידו drawn sword in his hand" shows up exactly two other places in Tanakh (Numbers 22:23 and Joshua 5:13) also seemingly unrelated to the Exodus.

Answer 1

• A weapon with a long blade for cutting or thrusting (source)
  Or this one (Avoda Zara 20b), a more fascinating description here

אמרו עליו על מלאך המות שכולו מלא עינים, בשעת פטירתו של חולה עומד מעל מראשותיו וחרבו שלופה בידו, וטיפה של מרה תלויה בו

As mentioned here, by Mida KeNeged Mida, Hashem punished the Egyptians with a sword since they thought to do that themselvs :
Sota 11a

א"ר חמא ברבי חנינא: באו ונחכם למושיען של ישראל במה נדונם נדונם באש? כתיב (ישעיהו סו טו) כי הנה ה' באש יבא [וכסופה מרכבתיו להשיב בחמה אפו וגערתו בלהבי אש], וכתיב כי באש ה' נשפט וגו' [ובחרבו את כל בשר ורבו חללי ה’] בחרב? כתיב 'ובחרבו את כל בשר'! אלא בואו ונדונם במים, שכבר נשבע הקב"ה שאינו מביא מבול לעולם שנאמר (ישעיהו נד ט) כי מי נח זאת לי וגו' [אשר נשבעתי מעבר מי נח עוד על הארץ כן נשבעתי מקצף עליך ומגער בך]

והיינו דא"ר אלעזר: מאי דכתיב (שמות יח יא) [עתה ידעתי כי גדול ה' מכל האלקים] כי בדבר אשר זדו עליהם בקדירה שבישלו בה נתבשלו.

When was this Plague of Sowrd ? HaShl"a (and probably more) explains it happened during the 10th plague, the firstborn insisted their fathers to release the isralites and upon refusal slayed them, with a sword.
Shl"a, Yefe Nof

• The refrence might be because it is a Pasuk from the Torah(didn't get that one, see there), and the Pasuk has the meaning of Hashem using a sword as a tool/mean of punishment.
  According to Yefe Nof, had hashem punished the firstborns by 'himself' that would have been a 'privilege', so we bring a Pasuk to show that it wasn't Hashem but the Sword of Malach Hamavet ( as it had been with David Hamelech)

Edit: Just now checked the Maharsh"a, the Pasuk in the gmara is the one from Bil'am (not David) and it proves that Hashem gives mida keneged mida ( to Bila'm who usually uses a sword and at that time switched to mouth-Kishuf). In addition to all of the above, we might want to keep a consistency and use the same expression (and have נטויה for Gzera Shava) to prove the prinicipal of Mida Keneged Mida.

Answer 2

The Baal Haggadah here is being subtle. Z'roa Netuya refers to the threat of a future violent act, just as a 'drawn sword' represents the threat of an attack, not the attack itself. And the possuk cited refers to a drawn sword seen in the skies over Yerushalayim, a future threat, not an attack.

I believe it is Netziv who asks why --out of frustration--Par'oh did not simply kill all of Bnei Yisrael in retaliation for the makkot. He answers that the makkot were Hashem's yad chazaka, but that there was another constant as well. To keep Par'oh off-balance Hashem arranged to kill off his ministers one by one. It is this aspect of haShem's hashgacha we call the Z'roa Netuya, and it kept Par'oh busy replacing those ministers. As a result of this ongoing (and future) threat to his government's stability, Par'oh had no time to act against Bnei Yisrael.

Answer 3

I can answer the second part of your question. The pasuk is quoted in order to make a "Gezairas Shava" between the two words "נְטוּיָה" in order to show that the word נְטוּיָה refers to a sword.

A "Gezairas Shava" is one of the exegesic principle by which the Torah is elucidated. The basic premise of Gezairas Shava is that the occurrence of the same word in two places comes to indicate a connection between the two places where that same word is mentioned. The exact details are beyond the scope of this answer, but this principle is extremely common in Medrash - tradition-based Biblical exegesis.

So in this case, we might wonder what exactly נְטוּיָה comes to indicate. We realize than another occurrence of נְטוּיָה is found in another Pasuk in Tanach to refer to a sword, and surmise that the first occurrence also refers to a sword. That is what the Haggada is trying to say.