The Scalar Triple Product Gives the Volume of a Parallelepiped
The Scalar Triple Product Gives the Volume of a Parallelepiped
In three dimensions, a parallelepiped is a prism whose faces are all parallelograms. Let , , and be the basis vectors defining a three-dimensional parallelepiped. Then its volume is given by the scalar triple product:
(,,)
x
1
y
1
z
1
(,,)
x
2
y
2
z
2
(,)
x
3
y
3,
z
3
V
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