Lord Śrī Rāma's weapon
Lord Śrī Rāma is the original divinity, the original personality of godhead, the two-armed Maha-Vishnu. All other Vishnu-Tattvas are his Sva-amshas (various manifestations with different names and forms). Lord Vishnu's prime weapon is Sudarshan chakra (disc) which is called by name Vishnu-chakra in Shri Valmiki Ramayana. Though, Bhagavan Shri Rama had also this Sudarshan Chakra - the disc of Vishnu in his Ramavatar, Yet he never used the Sudarshan-Chakra, as his nija-aayudh (own weapon) is his divine bow.
The original divinity is two-armed
In Veda-avatar Valmiki Ramayana, at the time of disappearing from the eyes of people, Bhagavan Shri Ram entered in his spiritual abode without discarding his prior form. This tells the Sva-Vigrah (own form) of Bhagavan Shri Ram is eternal and original.
In Vedas, there is enquiry about the two arms of Bhagavan in his personal form? e.g. ‘किं बाहू (यजुर्वेद ३०.१०) / कौ बाहू (ऋग्वेद १०.०९०.११)’ — ‘kiṃ bāhū (yajurvēda 30.10) / kau bāhū (ṛgvēda 10.090.11)’ — in bāhū (two arms ) there is Dvivachana (the dual number) [~ bāhu - arm; bāhū - two arms; bāhavaḥ - many arms].
Bhagavan Valmiki composed Shri Ramayana in order to reinforce the true imports of Vedas. Brahman is the ultimate aim of the Vedas, and Shri Rama is the one who is sung throughout in the beginning, the middle, the end of the Valmiki-Ramayana.
It is said in many hymns of Vedas that the Eko Advitiya Brahm, the Param Purusha (the Supreme personality of Godhead, who is second to none) wields bow and arrow in his hands. Thus, The two-armed form of Bhagavan Shri Rama is the eternal-form and the original primeval form.
The original divinity wields bow and arrow
नमस्ते रुद्र मन्यव उतो त इषवे नमः।
नमस्ते अस्तु धन्वने बाहुभ्याम् उत ते नमः॥
(कृष्ण-यजुर्वेद ४.५.१)
namastē rudra manyava utō ta iṣavē namaḥ।
namastē astu dhanvanē bāhubhyām uta tē namaḥ ॥
(kṛṣṇa-yajurvēda 4.5.1)
"O' Lord Shri Rama (Rudra)! Salutations to Your anger and salutations also to Your arrow. May our salutations be to Your bow and salutations to Your two hands (which wield the bow and the arrow).
→ The word 'Rudra' is derived from ‘rudaM saMsAraduHkhaM drAvayatIti rudra iti - the one who drives away the sorrows of the world’, ‘rOdayati nishAcharAn yah sa rudraH - the one who makes his enemies cry at the time of destruction in war (शत्रुओं को रुलाते हैं जो)’, ‘rutim dadAti iti rudrah - the one who gives Vedic-shabda (rutim) to BrahmA in the beginning of Kalpa’, thus by the word ‘Rudra’ here Bhagavan Shri Rama should be referred primarily.
“Each and every name in Vedas primarily connotes Shri Rama (the 2-armed Vishnu, The Purusha of Vedas). While Other individual deities are referred as secondary meanings only.”
→ Here again there is dvivachan in बाहुभ्याम् (bāhubhyām) - salutations for (to) two arms of Bhagavan in the Vaidic-hymn [~ बाहवे bāhavē - to arm; बाहुभ्याम् bāhubhyām - to two arms; बाहुभ्यः bāhubhyaḥ - to many arms]. The ‘two arms’ are indicative of the personal form (Narakruti) of Bhagavan Shri Rama, the two-armed form is the most superior as it comes for 'ekam advitiy Purush - one most superior Personality' in other Vaidic mantra.
→ Bow and arrow are the personal aayudh (weapons) of the Purusha of Vedas, Bhagavan Shri Rama.
→ Shri Rama’s bow and infallible arrows bestow assurance to devotees that He is the supreme protector and the sole refuge of entire beings!
→ Though Lord’s divine Rupa-Madhuri in itself is enough to entice entire cosmos by its beauty, Yet when he stands with his Bow and arrows it bestows a great bliss to devotees, It captures our mind with an allure to relish Lord’s heroic appearance!
करतल बान धनुष अति सोहा। देखत रूप चराचर मोहा॥
(श्रीरामचरितमानस १.२०४.४)
"With an arrow and bow in the hands, Shri Rama appears most charming and his beauty enraptures the whole creation, both animate and inanimate." (Shri Ramcharitmanas 1.204.4)
मरीचिमण्डले संस्थं बाणाद्यायुधलाञ्छितम्।
द्विहस्तमेकवक्त्रं च रूपमाद्यमिदं हरेः ॥
(पद्म-संहिता)
marīcimaṇḍalē saṃsthaṃ bāṇādyāyudhalāñchitam।
dvihastamēkavaktraṃ ca rūpamādyamidaṃ harēḥ ॥
(padma-saṃhitā)
"In the Sun (the divine light/effulgence), there is the original primeval-form of Shri Hari having one face and two hands with bow and arrow."
(Thus, two-armed and one face i.e. Manujakriti is called the original form of Shri Hari. and The original personality of Godhead has two arms, and he wields bow and arrow, therefore all Vedas sing the glory of Purusha with bow and arrow. As these are the personal weapons of Veda-Purusha (Shri Rama), all Vaishnavas irrespective of their ishta-svarupa can get it on their arms in Vaishnava-S it is the supreme fortune to receive the imprints of the divine bow and arrow of Shri Rama after the instruction and guidance of one's Gurudev.)
Shri Hari Vishnu is Upmaan (simile) and Shri Ram is Upmeya (to whom simile of Vishnu is given). Shri Ram uses his own weapons which are bow and arrow. Bow represents ॐ, the curve shaped bow is the incarnation of Lakshman Ji, and the straight arrow is incarnation of Hanuman Ji, Both of them are inseparable part of Bhagavan Shri Rama.
There is another reason why Shri Rama doesn't like to use Sudarshan-disc, It is because once Shri Vishnu hurled his Sudarshan-disc on Ravana, yet Sudarshan Chakra remained ineffective in front of Ravana and it failed after making some bruise on Ravana's arm.
Upanishad says Bow represents the sacred syllable 'Aum' and arrow represents 'soul'.
धनुर् गृहीत्वौपनिषदं महास्त्रं शरं ह्युपासा निशितं सन्धयीत।
आयम्य तद्भावगतेन चेतसा लक्ष्यं तदेवाक्षरं सोम्य विद्धि॥३॥
(मुण्डकोपनिषद २.२.३)
dhanur gṛhītvaupaniṣadaṃ mahāstraṃ śaraṃ hyupāsā niśitaṃ sandhayīta।
āyamya tadbhāvagatēna cētasā lakṣyaṃ tadēvākṣaraṃ sōmya viddhi॥3॥
(muṇḍakōpaniṣada 2.2.3)
Meaning: Take my friend, this bow, this great weapon of Upanisad; place veneration on it as the whetted arrow; stretch it with the thought fixed on the nature of that (brahman); that very imperishable is the target, my friend. Strike it! (Mundakopanisad 2.2.3)
प्रणवो धनु:शरोह्यात्मा ब्रह्मतल्लक्ष्यमुच्यते।
अप्रमत्तेन वेद्धव्यं शरवत्तन्मयो भवेत्॥४॥
(मुण्डकोपनिषद २.२.४)
praṇavō dhanu:śarōhyātmā brahmatallakṣyamucyatē।
apramattēna vēddhavyaṃ śaravattanmayō bhavēt॥4॥
(muṇḍakōpaniṣada 2.2.4)
Meaning: The Sacred syllable 'Aum' is called the bow, the arrow is the soul, and Brahman is its target; He shall be pierced by him whose attention does not swerve. Then he will become one with Him as the arrow becomes one with the target whence it has pierced it. (Mundakopanisad 2.2.4)
हिरण्मये परे कोशे विरजं ब्रह्म निष्कलम् ।
तच्छुभ्रं ज्योतिषं ज्योतिस्तद् यदात्मविदो विदुः॥९॥
(मुण्डकोपनिषद २.२.९)
Meaning: In the supreme bright sheath is Brahman (Shri Rama), free from taints and without parts. It is pure, and is the Light of lights. It is that which the knowers of the Self realize. (Mundakopanisad 2.2.9)
।।श्रीमद्रामचंद्रचरणौ शरणं प्रपद्ये।।
।। श्रीमते रामचन्द्राय नमः ||
।।śrīmadrāmacaṃdracaraṇau śaraṇaṃ prapadyē।।
।। śrīmatē rāmacandrāya namaḥ ||