在平面四边形ABCD中,|AC|=3,|BD|=2,则（AB+DC）(AC+BD)=

题目

在平面四边形ABCD中,|AC|=3,|BD|=2,则（AB+DC）(AC+BD)=
注：AC,BD,AB,DC都为向量

答案

→ → → → → → → → → →
AB + DC = AC + CB + DB + BC = AC + DB = AC - BD
→ → → →
(AB + DC)(AC + BD)
→ → → →
=(AC - BD)(AC + BD)
→ →
=(AC)²-(BD)²
→ →
=|AC|²-|BD|²
=3²-2²
=5