Question

已知a、b、c是平面内三个向量,若向量c满足(a-c)(b-c)=0,且|c|=1,|a|=|b|=2,则|a+b|的最

Answer 1

(a-c).(b-c)=0

a.b-(a+b).c+|c|^2=0

a.b = (a+b).c- |c|^2

= (a+b).c -1

max a.b = |a+b||c| -1 = |a+b|-1

|a+b|^2=|a|^2+|b|^2+2a.b

=8+2a.b

max |a+b|^2 when a.b is max

=>

max|a+b|^2 = 8 + 2(max|a+b|-1)

max|a+b|^2 -2max|a+b| -6=0

max|a+b|=1+√7