50 4 Epsilon-Tensor
4.1.3 Antisymmetric Tensor Linked with a Vector
With the help of the epsilon-tensor and its properties, the dual relation (4.3) which
links a vector with an antisymmetric second rank tensor can be inverted. This is
seen as follows. First, by renaming indices, (4.3) is rewritten asaλ=ελμ′ν′Aμ′ν′.
Multiplication of this expression byεμνλand use of (4.10) leads to
εμνλaλ=(δμμ′δνν′−δμν′δνμ′)Aμ′ν′=Aμν−Aνμ= 2 Aasyμν, (4.13)or
A
asy
μν=1
2
εμνλaλ. (4.14)4.2 Multiple Vector Products
4.2.1 Scalar Product of Two Vector Products
Leta,b,c,dbe four vectors. The scalar product of two vector product formed with
these vectors can be expressed in terms of products of scalar products:
(a×b)·(c×d)=a·cb·d−a·db·c. (4.15)For the proof, notice that the left hand side of (4.15) is equivalent to
ελμνaμbνελμ′ν′cμ′dν′.Use of the symmetry properties of the epsilon-tensor and of (4.10) allows one to
rewrite this expression as(δμμ′δνν′−δμν′δνμ′)aμbνcμ′dν′=aμcμbνdν−aμdμbνcν,
which corresponds to the right hand side of (4.15).
Now the special casec=a,d=bis considered. Then (4.15) implies
|a×b|^2 =a^2 b^2 −(a·b)^2 =a^2 b^2
(
1 −(̂a·̂b)^2)
=a^2 b^2 ( 1 −cos^2 φ)=a^2 b^2 sin^2 φ.
(4.16)
Herêaand̂bare unit vectors andφis the angle between the vectorsaandb.The
last equality is equivalent to (3.42).
4.2.2 Double Vector Products.
Leta,b,cbe three vectors. The double vector producta×(b×c)is a vector with
contributions parallel tobandc, in particular