Volume of Parallelepiped

A parallelepiped is a six-faced geometric figure (also known as a polyhedron) where each face is a parallelogram. It is a three-dimensional counterpart of a parallelogram in two dimensions. The volume of a parallelepiped is a measure of the space enclosed by its faces.

Understanding the Parallelepiped

A parallelepiped is defined by three vectors that emanate from the same point (corner) and represent the edges of the parallelepiped. These vectors are often denoted as a, b, and c. The volume of the parallelepiped can be calculated using the scalar triple product of these vectors.

Scalar Triple Product

The scalar triple product of vectors a, b, and c is given by the dot product of one of the vectors with the cross product of the other two. It is denoted as a · (b × c) and is a scalar quantity.

The formula for the scalar triple product is:

$$ V = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) $$

The absolute value of the scalar triple product gives the volume of the parallelepiped:

$$ |V| = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})| $$

Properties of the Scalar Triple Product

• The scalar triple product is positive if the vectors a, b, and c form a right-handed system, and negative if they form a left-handed system.
• The scalar triple product is zero if any two of the vectors are parallel or if all three are coplanar.
• The scalar triple product is invariant under a cyclic permutation of the vectors, meaning that a · (b × c) = b · (c × a) = c · (a × b).

Calculating the Volume

To calculate the volume of a parallelepiped, we can use the following steps:

1. Compute the cross product of two vectors, say b × c.
2. Take the dot product of the resulting vector with the third vector, a.
3. Take the absolute value of the result to ensure the volume is positive.

Example

Let's consider vectors a = (1, 2, 3), b = (4, 5, 6), and c = (7, 8, 9). The volume of the parallelepiped formed by these vectors is:

1. Compute b × c:

$$ \mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 4 & 5 & 6 \ 7 & 8 & 9 \ \end{vmatrix} = (5 \cdot 9 - 6 \cdot 8) \mathbf{i} - (4 \cdot 9 - 6 \cdot 7) \mathbf{j} + (4 \cdot 8 - 5 \cdot 7) \mathbf{k} = (-3, 6, -3) $$

1. Compute a · (b × c):

$$ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = (1, 2, 3) \cdot (-3, 6, -3) = 1 \cdot (-3) + 2 \cdot 6 + 3 \cdot (-3) = -3 + 12 - 9 = 0 $$

1. Since the scalar triple product is zero, the volume of the parallelepiped is also zero. This indicates that the vectors are coplanar.

Differences and Important Points

Property	Description
Definition	The volume of a parallelepiped is the scalar triple product of its defining vectors.
Formula	(
Geometric Interpretation	The volume represents the space enclosed by the parallelepiped.
Significance of Zero Volume	If the volume is zero, it indicates that the vectors are coplanar.
Cyclic Permutation	The scalar triple product is invariant under cyclic permutation of the vectors.

In summary, the volume of a parallelepiped is a fundamental concept in vector calculus and geometry, and it is widely used in physics and engineering to determine the magnitude of a parallelepiped-shaped region in space. Understanding how to calculate it using the scalar triple product is essential for solving problems related to three-dimensional space.