show that a cross bcrossc cross dca dot b cross dda dot b cr

show that

(a cross b)cross(c cross d)=c(a dot b cross d)-d(a dot b cross c)

in 3d

Solution

I\'m assuming you already know the proof of this theoram:

Equation1:

AxBxC = (A.B).C - (A.C).B

Equation 2:

(AxB).C = (BxC).A

(AxB)x(CxD) = C(A.BxD) - D(A.BxC)

In the Left hand side

Let AxB = E

Ex(CxD) = (E.D).C - (E.C).D

= C.(E.D) - D.(E.C)

put back E = AxB

= C.(AxB.D) - D(AxB.C)

= C[ABD] - D[ABC]

from equation 2

= C[BxD.A] - [BxC.A]D