va a1, a2, a3;
vb b1, b2, b3;
vc c1, c2, c3;
vd d1, d2, d3;
Some properties of the cross product and dot product
üMixed product a.(b × c)
The product a.(b x c) is called the mixed product. The geometric meaning of the mixed product is the volume of the
parallelepiped spanned by the vectors a, b, c, provided that they follow the right hand rule. This fact is consistent with
the above identities.
It turns out that the mixed product equals the 3×3 determinant whose rows are the coordinates of the vectors in the
order of the mixed product.
a.bcdet a1a2
b1b2
c1c2
a3
b3
c3
The most important property of the mixed product is that it is “circular”
a.(b × c) = c.(a × b) = b.(c × a)
Here is a Mathematica “proof” of this property
Simplifyva.Crossvb, vcvc.Crossva, vb
0
Simplifyva.Crossvb, vcvb.Crossvc, va
0
üDouble cross product a×(b×c) = (a.c) b - (a.b) c
The following identity holds for the double cross product of three vectors
a×(b×c) = (a.c) b - (a.b) c
Here is a Mathematica proof of this identity:
SimplifyCrossva, Crossvb, vc va.vcvb va.vbvc
0, 0, 0
How to prove this Identity?
üHere is a proof
First we prove the identity with the unit mutually orthogonal vectors u and v. Let u and v be unit vectors such that u
and v are orthogonal. Then
a×(u×v) = (a.v) u - (a.u) v
To prove this identity we notice that the vector a×(u×v) is orthogonal to u×v and therefore must be in the plane
spanned by u and v. Therefore a×(u×v) must be a linear combination of u and v. That is
a×(u×v) = a u + b v
Since u and v are unit vectors and u¦v, by the definition of the cross product we have (u×v)×u = v and (u×v)×v = -
u. Further, since u¦v, we have u.v = 0. Therefore
a = u.(a×(u×v)) = a.((u×v)×u) = a.v
