Download PDF

triple scalar product∗
slider142†
2013-03-21 12:50:32
The triple

a1 b1
det  a2 b2
a3 b3
scalar product of three vectors is defined in R3 as

c1
b2 c2
b1 c1
b1
~

c2 = ~a·(b×~c) = a1 det
−a2 det
+a3 det
b3 c3
b3 c3
b2
c3
The determinant above is positive if the three vectors satisfy the right-hand rule
and negative otherwise. Recall that the magnitude of the cross product of two
vectors is equivalent to the area of the parallelogram they form, and the dot
product is equivalent to the product of the projection of one vector onto another
with the length of the vector projected upon. Putting these two ideas together,
we can see that
|~a · (~b × ~c)| = |~b × ~c||~a| cos θ = base · height = Volume of parallelepiped
Thus, the magnitude of the triple scalar product is equivalent to the volume of
the parallelepiped formed by the three vectors. It follows that the triple scalar
product of three coplanar or collinear vectors is then 0. When we keep the sign
of the volume, we preserve orientation.
View SVGTripleScalarProduct.svg
Note that there is no rigid motion, barring reflection, that will allow you to
move/rotate the left parallelepiped so that the respective vectors coincide. This
is the sense of orientation that is used in differential geometry.
Identities related to the triple scalar product:
~ × B)
~ ·C
~ = (B
~ × C)
~ ·A
~ = (C
~ × A)
~ ·B
~
• (A
~ · (B
~ × C)
~ = −A
~ · (C
~ × B)
~
• A
∗ hTripleScalarProducti
created: h2013-03-21i by: hslider142i version: h30899i Privacy
setting: h1i hDefinitioni h47A05i h46M05i h11H46i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1
c1
c2
The latter is implied by the properties of the cross product.
Generalization:
Given a metrizable manifold M , the scalar n-product of an ordered list of n
vectors (v~1 , v~2 , . . . , v~n ) corresponds to the signed n-volume of the n-paralleletope
spanned by the vectors and is calculated by:
∗(
n
_
v~i )
i=1
where ∗ is the Hodge star operator and ∨ is the exterior product. Note that
this simply corresponds to the determinant of the list in n-dimensional Euclidean
space.
2