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1(c). Triple Products
Since the cross product of two vectors is itself a vector, it can be dotted or crossed with a
third vector to form a triple product.
(i) Scalar triple product:
.
A B C
Geometrically
.
A B C
is the volume of the parallelepiped
generated by
,
A B
and
C
, since
B C
is the area of the base,
and
cos
A
is the altitude. Evidently,
. . .
A B C B C A C A B
In component form
.
x y z
x y z
x y z
A A A
A B C B B B
C C C
Note that the dot and cross can be interchanged:
. .
A B C A B C
(ii) Vector triple product:
A B C
The vector triple product can be simplified by the so-called BAC-CAB rule:
. .
A B C B A C C A B
B
C
A
n
ˆ
