Question

a,b,c为任意方向向量,证明(a×b)×c=(a·c)·b-(b·c)·a

Answer 1

设：a向量=(x1,y1,z1) b向量=(x2,y2,z2) c向量=(x3,y3,z3)
(a×b)×c=(y1z2-y2z1,x2z1-x1z2,x1y2-x2y1)×(x3,y3,z3)
=[z3(x2z1-x1z2)-y3(x1y2-x2y1),
x3(x1y2-x2y1)-z3(y1z2-y2z1),
y3(y1z2-y2z1)-x3(x2z1-x1z2)]
(a·c)·b-(b·c)·a =(x1x3+y1y3+z1z3)[x2,y2,z2]
-(x2x3+y2y3+z2z3)[x1,y1,z1]
=[(x1x3+y1y3+z1z3)x2-(x2x3+y2y3+z2z3)x1,
(x1x3+y1y3+z1z3)y2-(x2x3+y2y3+z2z3)y1,
(x1x3+y1y3+z1z3)z2-(x2x3+y2y3+z2z3)z1]
上下两个化简后比较下,可得两者相等