42 3 Symmetry of Second Rank Tensors, Cross Product
To verify this expression, e.g. useμ=1 and note thatδ 11 =1,δ 21 =δ 31 =0. Then
one obtains the first line of (3.36). Similarly, the second and third line are recovered
withμ=2 andμ=3.
The cross product of the two vectors is antisymmetric with respect their exchange:
(b×a)=−(a×b). (3.39)When the two vectors are parallel, i.e. when one hasb=kawith some numerical
factork, the cross product is equal to zero, thus
a×b= 0 ⇐⇒a‖b. (3.40)These properties follow from the definition of the vector product. Likewise, with the
help of (3.38), the scalar product of a vectordwith the vectorcwhich, in turn, is the
vector product of vectorsaandbis given by thespate product:
d·c=d·(a×b)=∣ ∣ ∣ ∣ ∣ ∣
d 1 a 1 b 1
d 2 a 2 b 2
d 3 a 3 b 3∣ ∣ ∣ ∣ ∣ ∣
=
∣ ∣ ∣ ∣ ∣ ∣
d 1 d 2 d 3
a 1 a 2 a 3
b 1 b 2 b 3∣ ∣ ∣ ∣ ∣ ∣
. (3.41)
Two vectorsdandcare orthogonal, when the scalar productd·cvanishes. From
(3.41) follows that the spate productsa·(a×b)andb·(a×b)are zero, since a
determinant with two equal columns or rows is zero. Thus the vector producta×bof
two vectorsaandbis perpendicular to bothaandb. Of course,aandbare assumed
not to be parallel to each other.
In summary, the vector product of two vectors which span a plane is defined such
thata×bis perpendicular to this plane. The direction of this vector is parallel to the
middle finger of theright handwhenapoints along the thumb andbis parallel to
the pointing finger. The magnitude of the vector product is given by the magnitude
ofatimes the magnitude ofbtimes the magnitude of the sine of the angleφbetween
aandb, viz.:
|a×b|=|a||b||sinφ|. (3.42)
For the proof of (3.42), see Fig.3.1. The coordinate system is chosen such thatais
parallel to the 1-axis andbis in the 1–2-plane, their components then are(a 1 , 0 , 0 )
and(b 1 ,b 2 , 0 ), witha 1 =a,b 1 =bcosφ,b 2 =bsinφ,a=|a|andb=|b|.
Due to (3.36), the components ofc=a×bare( 0 , 0 ,c 3 ), withc 3 =a 1 b 2 =
absinφ. The magnitude ofc 3 then is given by (3.42). The magnitude of the vector
product assumes it maximum value when the two vectors are orthogonal.