Contents

The **volume** of a **parallelepiped** serves as an intriguing point of exploration, while embarking on a journey into the realm of **three-dimensional space**.

As a **polyhedron** enveloped by six **parallelograms**, a **parallelepiped** is a geometric marvel that offers rich insights into the interplay of **vectors** and spatial dimensions.

This article aims to unfold the** intricacies** of **parallelepipeds**, diving into the concept, its intriguing properties, and the** mathematical elegance** of its **volume calculation**.

**Strap** in as we traverse the **vibrant landscape** of** parallelepipeds**, delving into a world where **geometry** coalesces with **algebra**, illuminating corners of mathematical understanding with fascinating clarity.

## Defining the Volume of a Parallelepiped

The **volume** of a **parallelepiped** is the measure of the **three-dimensional space** it encompasses or occupies. In terms of **vectors**, if a **parallelepiped** is formed by three vectors **a**, **b**, and **c,** in three-dimensional space starting from the same point, the** volume** is calculated using the **scalar triple product** of these vectors.

Mathematically, this is represented as the **absolute value** of the **dot product** of vector **a** and the **cross product** of vectors **b** and **c**, denoted as **V = |a . (b x c)|**. This volume calculation is a reflection of the **parallelepiped’s spatial properties**, taking into account the lengths of its edges and the angles between them.

Below in figure-1, we present a generic diagram for a parallelepiped with its volume.

Figure-1.

**Computing the Volume of a Parallelepiped**

The **volume (V)** of a **parallelepiped** can be found using the **scalar triple product** of the three vectors defining the edges of the **parallelepiped**. If vectors a, b, and c form the edges of the parallelepiped, the volume is given by:

V = | a . (b x c) |

Where:

**“.”**denotes the**dot product**of two**vectors**.**“x”**denotes the**cross product**of two**vectors**.**“|”**around the expression denotes the**absolute value**.

The **scalar triple product** is equivalent to the **determinant** of a **3×3** **matrix** with the components of vectors **a**, **b**, and** c** as its **rows** or **columns**:

V = | det([a; b; c]) |

It’s important to note that the** volume of a parallelepiped** is always **positive**, so the **absolute value operation** ensures this.

## Properties

The **volume of a parallelepiped**, a **three-dimensional geometric** entity characterized by** six parallelogram** faces, has several mathematical and geometrical defining properties. Understanding these properties can provide a profound insight into three-dimensional space and its **geometric manifestations**.

**Defined by Scalar Triple Product**

One of the central properties of the **volume** of a parallelepiped is that it is given by the **scalar triple product** of three vectors **a**, **b**, and **c** that define the edges of the parallelepiped. The scalar triple product of **a**, **b**, and **c** is calculated as the **absolute value** of vector **a’s dot product** and the **cross product** of vectors **b** and **c**, denoted as **V = |a . (b x c)|**.

**Non-negative Quantity**

The **volume** of a **parallelepiped i**s always a **non-negative** quantity. This is because it represents a **physical quantity**, the amount of space occupied by the parallelepiped, which cannot be negative. The **scalar triple product’s absolute value** ensures the volume’s **non-negativity**.

**Zero Volume Implies Coplanar Vectors**

If the volume of a **parallelepiped** is **zero**, it implies that the three vectors defining the edges of the **parallelepiped** are **coplanar**, i.e., they lie in the same **plane**. This is because the volume, computed as the** scalar triple product**, will be zero if the vectors are **coplanar**, as the height of the **parallelepiped** would be zero in such a case.

**Invariant under Permutations of Vectors**

The **volume** of the **parallelepiped** remains the same even if the order of the vectors **a**, **b**, and **c** in the scalar triple product is **permuted ****cyclically**, i.e., **V = |b . (c x a)| = |c . (a x b)|**. This is because the **cyclic permutation** of the vectors does not change the **physical configuration** of the **parallelepiped**.

### Change of Sign Under Anti-cyclic Permutations

The **volume** changes sign under an **anti-cyclic permutation** of the vectors **a**, **b**, and **c**, i.e., **V = – |a . (c x b)|**. Although the volume itself, being an absolute value, is always **non-negative**, the scalar triple product can be **negative**, reflecting the vectors’ orientation.

**Dependence on Edge Lengths and Angles**

The **parallelepiped** volume depends on the **edges’ lengths** and the **angles** between them. More specifically, it’s the product of the **areas of the base** (given by the magnitude of the **cross product** of vectors **b** and **c**) and the **height** (given by the **projection** of a vector **a** onto the vector **perpendicular** to the base).

**Connection to Determinants**

The **scalar triple product** that gives the volume of a parallelepiped can also be viewed as the **determinant** of a **3×3 matrix** whose rows or columns are the components of the vectors **a**, **b**, and **c**. This links the volume of a parallelepiped and the determinant concept in **linear algebra**.

**Applications **

### Mathematics

In **mathematics**, the **volume** of a **parallelepiped** is an important concept in **three-dimensional geometry**. It is used to calculate the volume of **irregularly shaped objects** and is a key component in the study of **solid geometry**.

### Physics

In **physics**, the **volume** of a **parallelepiped** is used to calculate the volume of **three-dimensional objects**, such as **containers**, **tanks**, or any other physical systems with a parallelepiped shape. It is an essential parameter in various physical calculations involving **mass**, **density**, **fluid flow**, and **material properties**.

### Engineering

In engineering disciplines, the **volume** of a **parallelepiped** is crucial for determining the **capacity**, **flow rate**, and **storage requirements** of **containers**, **pipes**, and **channels**. It is also used in **structural analysis** to calculate **solid objects’ displacement**, **stress**, and **strain**.

### Architecture

In **architecture**, the **volume** of a **parallelepiped** is used to measure the enclosed space within a **building** or **room**. It is essential for determining room dimensions, and material quantities, and estimating costs. Additionally, it plays a role in designing efficient ventilation and **heating/cooling systems**.

### Computer Graphics and Animation

In **computer graphics** and **animation**, the volume of a **parallelepiped** is used to define the **boundaries** and **physical characteristics** of **3D objects**. It is vital for creating **realistic simulations**,** rendering scenes**, and **modeling** complex shapes in **virtual** environments.

### Manufacturing and Material Science

In **manufacturing processes**, the volume of a **parallelepiped** is used to calculate** material requirements**, determine material **utilization rates**, and **estimate production costs**. It is also relevant in material science for **analyzing** properties such as **density**, **porosity**, and** elasticity**.

### Fluid Dynamics

In **fluid dynamics**, the volume of a **parallelepiped** is used to calculate the volume of **fluid displaced** by an object **immersed** in a fluid. This information is crucial for understanding **buoyancy** forces, **hydrostatic pressure**, and **fluid flow** characteristics.

**Exercise **

**Example 1**

Given vectors **a = [2, 3, 4]**, **b = [1, 1, 1]**, and** c = [0, 2, 3]**, calculate the **volume of the parallelepiped** spanned by these vectors.

**Solution**

The volume **V** of a **parallelepiped** can be found using the** scalar triple product** of the three vectors. So:

V = |a . (b x c)|

First, we calculate the **cross product** of vectors b and c:

b x c = [(1)(3) – (1)(2), (1)(0) – (1)(3), (1)(2) – (1)(0)]

b x c = [1, -3, 2]

Then, calculate the **dot product** of vector a and the result:

a . (b x c) = (2)(1) + (3)(-3) + (4)(2)

a . (b x c) = 2 – 9 + 8

a . (b x c) = 1

Taking the absolute value gives us the **volume of the parallelepiped**:

V = |1| = 1

**Example 2**

Given vectors **a = [4, 1, -1]**, **b = [2, 0, 2]**, and **c = [1, 1, 1]**, find the **volume of the parallelepiped** spanned by these vectors.

**Solution**

Calculate the volume using the **scalar triple product**:

V = |a . (b x c)|

First, find the **cross product** **b x c**:

b x c = [(0)(1) – (2)(1), (2)(1) – (2)(1), (2)(1) – (0)(0)]

b x c = [-2, 0, 2]

Then, calculate the **dot product** with vector **a**:

a . (b x c) = (4)(-2) + (1)(0) + (-1)(2)

a . (b x c) = -8 – 2

a . (b x c) = -10

The **volume of the parallelepiped** is the absolute value of this result:

V = |-10| = 10

Figure-2.

**Example 3**

Given vectors **a = [3, 0, 0]**, **b = [0, 3, 0]**, and **c = [0, 0, 3]**, calculate the **volume of the parallelepiped** spanned by these vectors.

**Solution**

Calculate the volume using the **scalar triple product**:

V = |a . (b x c)|

First, calculate the **cross product** **b x c**:

b x c = [(0)(3) – (0)(3), (3)(0) – (0)(3), (0)(3) – (0)(0)]

b x c = [0, 0, 9]

The **dot product** of vector a and the result is then:

a . (b x c) = (3)(0) + (0)(0) + (0)(9)

a . (b x c) = 0

So, the** volume of the parallelepiped** is:

V = |0| = 0

The vectors are **coplanar**.

Figure-3.

**Example 4**

Given vectors **a = [2, 2, 2]**, **b = [1, 1, 1]**, and **c = [3, 3, 3]**, find the **volume of the parallelepiped** spanned by these vectors.

**Solution**

Calculate the volume using the **scalar triple product**:

V = |a . (b x c)|

First, find the** cross product** **b x c**:

b x c = [(1)(3) – (1)(3), (1)(3) – (1)(3), (1)(3) – (1)(3)]

b x c = [0, 0, 0]

The **dot product** of vector a and the result is then zero, because the **cross product** is** zero vector**:

a . (b x c) = (2)(0) + (2)(0) + (2)(0)

a . (b x c) = 0

The **volume of the parallelepiped** is the absolute value of this result:

V = |0| = 0

The vectors are **coplanar**.

**Example 5**

Given vectors **a = [-1, 2, -3]**, **b = [4, -5, 6]**, and **c = [-7, 8, -9]**, find the **volume of the parallelepiped** spanned by these vectors.

**Solution**

Calculate the volume using the **scalar triple product**:

V = |a . (b x c)|

First, find the **cross product** **b x c**:

b x c = [(-5)(-9) – (6)(8), (6)(-7) – (4)(-9), (4)(8) – (-5)(-7)]

b x c = [-3, 6, -3]

The **dot product** of vector a and the result is:

a . (b x c) = (-1)(-3) + (2)(6) + (-3)(-3)

a . (b x c) = 3 + 12 + 9

a . (b x c) = 24

The **volume of the parallelepiped** is the absolute value of this result:

V = |24| = 24

**Example 6**

Given vectors **a = [1, 0, 2]**, **b = [-1, 2, 1]**, and **c = [0, 1, 1]**, calculate the **volume of the parallelepiped** spanned by these vectors.

**Solution**

Calculate the volume using the **scalar triple product**:

V = |a . (b x c)|

First, calculate the **cross product b x c**:

b x c = [(2)(1) – (1)(1), (1)(0) – (-1)(1), (-1)(1) – (2)(0)]

b x c = [1, 1, -1]

The **dot product** of vector a and the result is then:

a . (b x c) = (1)(1) + (0)(1) + (2)(-1)

a . (b x c) = 1 – 2

a . (b x c) = -1

The **volume of the parallelepiped** is the absolute value of this result:

V = |-1| = 1

*All images were created with MATLAB.*