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Scalar Triple Product
The
Scalar Triple Product
of 3
3-Dimensional Vectors
→
A
,
→
B
and
→
C
is calculated as follows
→
A
⋅
(
→
B
×
→
C
)
=
[
→
A
→
B
→
C
]
=
|
A
x
A
y
A
z
B
x
B
y
B
z
C
x
C
y
C
z
|
=
|
A
x
B
x
C
x
A
y
B
y
C
y
A
z
B
z
C
z
|
where
→
A
=
A
x
ˆ
i
+
A
y
ˆ
j
+
A
z
ˆ
k
→
B
=
B
x
ˆ
i
+
B
y
ˆ
j
+
B
z
ˆ
k
→
C
=
C
x
ˆ
i
+
C
y
ˆ
j
+
C
z
ˆ
k
Following are
Properties of Scalar Triple Product
The
Scalar Triple Product
of 3 Vectors gives the
Signed Volume of Parallelopiped Bound by the 3 Vectors
The Value of the Scalar Triple Product Does Not Change if the Dot and the Cross Products are Interchanged
as shown below
→
A
⋅
(
→
B
×
→
C
)
=
(
→
A
×
→
B
)
⋅
→
C
⇒
[
→
A
→
B
→
C
]
=
[
→
C
→
A
→
B
]
The Value of the Scalar Triple Product Does Not Change on Cyclic Permutation of the Vectors
as shown below
→
A
⋅
(
→
B
×
→
C
)
=
→
C
⋅
(
→
A
×
→
B
)
=
→
B
⋅
(
→
C
×
→
A
)
⇒
[
→
A
→
B
→
C
]
=
[
→
C
→
A
→
B
]
=
[
→
B
→
C
→
A
]
The Sign of Value of the Scalar Triple Product Changes on Non-Cyclic Permutation of the Vectors
as shown below
→
A
⋅
(
→
B
×
→
C
)
=
−
→
A
⋅
(
→
C
×
→
B
)
=
−
→
C
⋅
(
→
B
×
→
A
)
=
−
→
B
⋅
(
→
A
×
→
C
)
⇒
[
→
A
→
B
→
C
]
=
−
[
→
A
→
C
→
B
]
=
−
[
→
C
→
B
→
A
]
=
−
[
→
B
→
A
→
C
]
The Value of Scalar Triple Product is 0 if any 2 Vectors are Parallel or Same, or if the 3 Vectors are Co-Planar (i.e. they lie on the same Plane)
Related Topics
Vector Triple Product
,
Scalar Quad Product
,
Vector Quad Product
,
Introduction to Vector Algebra
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