Question

在△ABC中,∠BAC=90,∠ABC=60 AD⊥BC于D,若向量AD

Answer 1

AD=xAB+yAC,则：x+y=1

|BD|/|DC|=y/x=1/3,故：x=3y

即：y=1/4,x=3/4

即：(x,y)=(3/4,1/4)

-------------------要证明的话：

BD=AD-AB,DC=AC-AD

BD、DC共线,即：BD=kDC

即：AD-AB=k(AC-AD)

即：(k+1)AD=AB+kAC,即：AD=AB/(k+1)+kAC/(k+1)

即：x=1/(k+1),y=k/(k+1),即：x+y=1

BD=AD-AB=(x-1)AB+yAC=-yAB+yAC=yBC

DC=AC-AD=-xAB+(1-y)AC=-xAB+xAC=xBC

故：|BD|/|DC|=y/x