can you help me to divide this sentence into several parts which can help me understand how this sentence is made up and figure out its accurate meaning as well?

Vedi una traduzione

＠dapao114514

Rnのベクトル　v,wに対して、内積を　ｖ・ｗで表すことにする、　

【Rnのベクトル】　v, w に　たいして、　【内積】 を　ｖ・ｗ　で　あらわす　こと　に　する

(1)
Above sentence just says definition of 【内積】（= inner product）using v and w vectors.
v = 【Rnのベクトル】= vector Rn
w = 【Rnのベクトル】= vector Rn
ｖ・ｗ = 【内積】= inner product of 2 vectors v and w.

(2)
【Rnのベクトル】　v, w に　たいして、　
In this specific case only, you can translate above to,
“ Using v, w that are vector Rn, or “For v, w that are vector Rn,

Rn = n-dimensional vector R

(3)
【内積】 を　v･w　で　あらわす
Express ( or define) “inner product” by v･w

　あらわす　＝　express ( define, in this case)
で　＝　by

Please note that "n" in Rn is at the right upper corner of R. ( I couldn't make the letter position change)