Question

在△ABC中,CA=(a1,a2),CB=(b1,b2),试用a1a2b1b2表示S△ABC

CA CB都为向量

Answer 1

CA=(a1,a2),CB=(b1,b2)
cosC=(CA·CB)/(|CA|*|CB|)
=(a1b1+a2b2)/[√(a1²+a2²)(b1²+b2²)]
sinC=√(1-cos²C)=√{1-(a1b1+a2b2)²/[(a1²+a2²)(b1²+b2²)]}
=√{[(a1²+a2²)(b1²+b2²)-(a1b1+a2b2)²]/[(a1²+a2²)(b1²+b2²)]}
=√{(a1²b2²+a2²b1²-2a1a2b1b2)/[(a1²+a2²)(b1²+b2²)]}
=|a1b2-a2b1|/(|CB|*|CA|)
∴SΔABC=1/2|CB|*|CA|sinC
=|a1b2-a2b1|/2