Question

已知A(2,1),B(3.2),D(-1,4)（1）证AB向量垂直AD向量（2）若ABCD为矩形，确定C坐标

（3）M为OD上动点，当向量MA*向量MB取最小值，求向量OM坐标

Answer 1

(1)
AB = OB-OA= (1,1)
AD = OD-OA = (-3,3)
AB.AD =(1,1).(-3,3) = 0
=> AB垂直AD
(2)
let C(x',y')
AB = DC
(1,1) = (3-x',2-y')
x' = 2 and y' = 1
C(2,1)
(3)
let M(x,y)
OM = kOD
(x,y) = (-k,4k)
S= MA.MB
= (OA-OM).(OB-OM)
=(2+k, 1-4k).(3+k, 2-4k)
=(2+k)(3+k)+ (1-4k)(2-4k)
= 6 + 5k+k^2 +2-12k+16k^2
=17k^2-7k+8
S' = 34k-7 =0
k = 7/34
S''=34>0 (min)
M = (-k,4k) = (-7/34, 14/17)