WOLFRAM|DEMONSTRATIONS PROJECT

The Scalar Triple Product Gives the Volume of a Parallelepiped

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x
1
y
1
z
1
x
2
y
2
z
2
x
3
y
3
z
3
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 (1.50, 0, 0) · ( (0, 1.50, 0)  (0, 0, 1.50) )  = 3.38
In three dimensions, a parallelepiped is a prism whose faces are all parallelograms. Let
(
x
1
,
y
1
,
z
1
)
,
(
x
2
,
y
2
,
z
2
)
, and
(
x
3
,
y
3,
z
3
)
be the basis vectors defining a three-dimensional parallelepiped. Then its volume
V
is given by the scalar triple product:
V=(
x
1
,
y
1
,
z
1
)·[(
x
2
,
y
2
,
z
2
)(
x
3
,
y
3
,
z
3
)]​​=(
x
2
,
y
2
,
z
2
)·[(
x
1
,
y
1
,
z
1
)(
x
3
,
y
3
,
z
3
)]​​=(
x
3
,
y
3
,
z
3
)·[(
x
1
,
y
1
,
z
1
)(
x
2
,
y
2
,
z
2
)]