已知三点A(x1,y1,z1)、B(x2,y2,z2)、C(x3,y3,z3)，求平面方程.
向量AB=向量OB-向量OA=(x2-x1,y2-y1,z2-z1)
向量AC=向量OC-向量OA=(x3-x1,y3-y1,z3-z1)
向量AB×向量AC=([y1z2-y1z3-y2z1+y2z3+y3z1-y3z2],[-x1z2+x1z3+x2z1-x2z3-x3z1+x3z2],[x1y2-x1y3-x2y1+x2y3+x3y1-x3y2])
即a=y1z2-y1z3-y2z1+y2z3+y3z1-y3z2,
b=-x1z2+x1z3+x2z1-x2z3-x3z1+x3z2,
c=x1y2-x1y3-x2y1+x2y3+x3y1-x3y2
带入(x1,y1,z1),
=>(y1z2-y1z3-y2z1+y2z3+y3z1-y3z2)x1+(-x1z2+x1z3+x2z1-x2z3-x3z1+x3z2)y1+(x1y2-x1y3-x2y1+x2y3+x3y1-x3y2)z1+d=0,
=>x1y1z2-x1y1z3-x1y2z1+x1y2z3+x1y3z1-x1y3z2-x1y1z2+x1y1z3+x2y1z1-x2y1z3-x3y1z1+x3y1z2+x1y2z1-x1y3z1-x2y1z1+x2y3z1+x3y1z1-x3y2z1+d=0,
=>x1y2z3-x1y3z2-x2y1z3+x2y3z1+x3y1z2-x3y2z1+d=0,
=>d=-x1y2z3+x1y3z2+x2y1z3-x2y3z1-x3y1z2+x3y2z1
原文：http://www.cnblogs.com/leejxyz/p/5238029.html