PUMaC 2023 · 代数（B 组） · 第 32 题
PUMaC 2023 — Algebra (Division B) — Problem 32
题目详情
- Find ( bf − ce ) + ( cd − af ) + ( ae − bd ) . 3 x 5 x
解析
- Find ( bf − ce ) + ( cd − af ) + ( ae − bd ) . Proposed by Sunay Joshi Answer: 54 3 2 2 2 Solution: Let u = ( a, b, c ), v = ( d, e, f ) be vectors in R . Then the identity | u × v | = | u | | v | − 2 2 2 2 2 2 2 2 ( u · v ) implies that the desired expression is simply ( a + b + c )( d + e + f ) − ( ad + be + cf ) . 2 This evaluates to 14 · 77 − 32 = 54. 3 x 5 x