Question

【20分】[高中向量]向量a,b,x,y满足关系：a=y-x,b=2x-y,|a|=|b|=1,a*b=0,则|x|+|

Answer 1

设x=(X1,X2),y=(Y1,Y2).[x,y是向量,(X1,X2),(Y1,Y2)都是坐标]

则a=y-x=(Y1-X1,Y2-X2),b=2x-y=(2X1-Y1,2X2-Y2)

∵|a|=|b|=1,即a＾2=b＾2=1

∴a＾2=(Y1-X1,Y2-X2)＾2=1,即 Y1＾2+X1＾2-2X1Y1+Y2＾2+X2＾2-2X2Y2=1①

b＾2=(2X1-Y1,2X2-Y2)＾2=1即4X1＾2+Y1＾2-4X1Y1+4X2＾2+Y2＾2-4X2Y2=1②

又∵a*b=0 即(Y1-X1,Y2-X2)*(2X1-Y1,2X2-Y2)=0

∴化简得-2X1＾2-Y1＾2+3X1Y1-2X2＾2-Y2＾2+3X2Y2=0③

①*3+③*2得:(Y1＾2+Y2＾2)-(X1＾2+X2＾2)=3④

②*3+③*4得:4(X1＾2+X2＾2)-(Y1＾2+Y2＾2)=3⑤

解方程组④⑤,得X1＾2+X2＾2=2

Y1＾2+Y2＾2=5

∴|x|+|y|=二次根号项(X1＾2+X2＾2)+二次根号项(Y1＾2+Y2＾2)

=根号2+根号5

(写在纸上比较好看懂,这上面不知道怎么打符号)