Question

Se dau punctele A (m+1,1),B (2,3),C(4,-m).
Sa se determine m din R știind ca AB*AC=OA*OB-2; AB, AC, OA, OB sunt vectori

Answer 1

AB=(xB-Xa)i+(yB-yA)j=(2-m-1)i+(3-1)j=(1-m)i+2j
AC=(xC-xA)i+(yC-yA)j=(4-m-1)i+(-m-1)j=(3-m)i+(-m-1)j
Stim ca i*i=i la patrat=j*j= |i|=|j|=1 si ca i*j=0
AB*AC=(1-m)i*(3-m)i+(1-m)i*(-m-1)j+2j*(3-m)i+2j*(-m-1)j=(1-m)(3-m)+2 (3-m)=3-m-3m+m la patrat+6-2m=m la patrat-6m+9
OA=xAi+yAj=(m+1)i+j
OB=xBi+yBi=2i+3j
OA*OB=(m+1)i*2i+(m+1)i*3j+j*2i+j*3j=2m+2+3=2m+5
m la patrat-6m+9=2m+5
m la patrat-6m+9-2m-5=0
m la patrat-8m+4=0
Si de aici rezolvi cu delta