Question

已知向量ON=(2,1),向量OA=(1,7),向量OB=(5,1),O为原点,M为直线ON上一点.

Answer 1

设,向量OM坐标为(X,Y),

令,OM=m*OP,(m为实数),

(X,Y)=m(2,1),

x=2m,y=m,则有X=2Y.

向量MA=OA-OM=(1-X,7-Y)=(1-2Y,7-Y),

向量MB=OB-OM=(5-X,1-Y)=(5-2Y,1-Y),

向量MA*MB=(1-2Y)(5-2Y)+(7-Y)(1-Y)

=4Y^2-12Y+5+(Y^2-8Y+7)

=5Y^2-20Y+12

=5(Y-2)^2-8,

当Y=2时,向量MA*MB取最小值,

X=2Y=4.

向量OM=(4,2)

向量MA*MB=-8,

向量MA=(-3,5),向量MB=(1,-1)

|MA|=√34,|MB|=√2.

cosAMB=MA*MB/|MA|*|MB|=-4/√17=-4√17/17.