Contents

In this article, we’ll demystify the complexity of the **scalar triple product**, unraveling its intriguing **mathematical structure**, **real-world applications**, and the exciting **pathways** it opens in understanding the **three-dimensional world** around us. Buckle up and join us on this **mathematical adventure**!

## Definition of Scalar Triple Product

The **scalar triple product** is a **mathematical operation** involving three vectors in **three-dimensional space**. It is represented as [**a** · (**b** x **c**)], where ‘**a**‘, ‘**b**,’ and ‘**c**‘ are vectors, ‘**·**‘ represents the **dot product**, and ‘**x**‘ denotes the **cross product**. This operation yields a **scalar** (a single number) rather than a vector, hence its name.

More concretely, the scalar triple product gives the **parallelepiped volume** (a three-dimensional shape with six faces, each a parallelogram) defined by the three vectors ‘**a**,’ ‘**b**,’ and ‘**c**.’ It can also determine whether the three vectors are **coplanar**: if the scalar triple product equals zero, the vectors lie in the same** plane**.

**Properties**

The **scalar triple product** has several important properties reflecting its unique role within **vector calculus**. These include:

**Scalar Value**

As the name suggests, the **scalar triple product** produces a **scalar** (a single real number) rather than a vector. This is because it combines the **dot product** (which gives a scalar) and the **cross product** (which gives a vector). The final operation is a dot product, yielding a **scalar** result.

**The Volume of a Parallelepiped**

The **scalar triple product’s absolute value** gives the **parallelepiped volume** formed by vectors **a**, **b**, and **c**. This is a **direct geometric interpretation** of the scalar triple product.

**Test for Coplanarity**

If the **scalar triple product** of three vectors is zero, the vectors are **coplanar**, i.e., all lie in the same plane. This is because the volume of the parallelepiped they would form is zero, indicating no “height” dimension and the vectors are confined to **two dimensions**.

**Permutation Property**

The **scalar triple product** follows a **cyclic permutation property**. This means that if the order of the vectors is cyclically permuted (**a, b, c** to **b, c, a** or **c, a, b**), the value of the scalar triple product remains unchanged.

**Change of Sign**

If the order of any two **vectors** is swapped (not in a cyclic way), the scalar triple product changes its sign. So, for instance, **a · (b x c) = – b · (a x c)**.

**Distributivity**

The **scalar triple product** is **distributive** over vector** addition**. This means that if a fourth vector d is introduced, then **a · [(b+c) x d] = a · (b x d) + a · (c x d).**

**Scalar Multiplicity**

If a **vector** in the **scalar triple product** is multiplied by a scalar, the result of the **scalar triple product** is multiplied by the same scalar. For instance, if **k** is a scalar, then (**ka**) · (**b x c**) = **k** (**a** · (**b x c**)).

These properties make the scalar triple product a flexible and powerful tool in many areas of **mathematics** and **physics**, including determining **orthogonality** and **parallelism** of vectors, solving systems of linear equations, and analyzing **geometric** and **physical** problems.

**Ralevent Formulas **

The **scalar triple product** is an important concept in **vector calculus** and has a few closely associated formulas. Let’s take a look at these:

**Basic Formula**

The **scalar triple product** of three vectors **a**, **b**, and **c** is given by **a · (b x c)**. Here, ‘**·**‘ is the **dot product**, and ‘**x**‘ is the **cross product**.

**Determinant Representation**

The **scalar triple product** can also be represented as the **determinant** of a **3×3 matrix** where the components of vectors **a**, **b**, and **c** form the matrix’s rows (or columns). If **a = [a1, a2, a3]**, **b = [b1, b2, b3]**, and **c = [c1, c2, c3]**, then the **scalar triple product** is given by the **determinant** of the following matrix:

| a1 a2 a3 |

| b1 b2 b3 |

| c1 c2 c3 |

**The volume of a Parallelepiped**

The **scalar triple product’s absolute value** equals the **parallelepiped volume** formed by vectors **a**, **b**, and **c**, represented by **|a · (b x c)|**.

**Coplanarity Check**

If **a · (b x c) = 0**, vectors **a**, **b**, and **c** are **coplanar**.

**Distributive Law**

The **scalar triple product** obeys a **distributive law**, similar to other arithmetic operations. If **d** is a fourth vector, then **a · [(b+c) x d] = a · (b x d) + a · (c x d)**.

**Scalar Multiplicity**

If one of the **vectors** is multiplied by a scalar **k**, the scalar triple product also gets multiplied by the same **scalar**. For example, if **a = k * u** for some vector **u**, then **(ka) · (b x c) = k (a · (b x c))**.

These formulas encapsulate the essence of the **scalar triple product** and provide a **robust** **mathematical foundation** for its various applications in fields like **physics**, **computer graphics**, **engineering**, and more.

## Evaluation Method of Scalar Triple Product

The **scalar triple product** of three vectors is calculated in two primary steps: first, you take the **cross product** of two vectors, and then the **dot product** of the resulting vector with the third vector. Here’s a detailed procedure using the vectors **a**, **b**, and **c**:

**Step 1**

**Calculate the Cross Product of Vectors b and c**

The **cross product** of two vectors, **b = [b1, b2, b3]** and **c = [c1, c2, c3]**, is given by another** vector**, calculated as follows:

b x c = [(b2 c3 – b3 c2), (b3 c1 – b1 c3), (b1 c2 – b2 c1)]

**Step 2**

**Calculate the Dot Product of Vector a and the Result from Step 1**

Next, calculate the **dot product** of vector **a = [a1, a2, a3]** and the resulting vector from the **cross product** **(b x c)** in Step 1.

a · (b x c) = a1 * (b1 x c1) + a2 * (b2 x c2) + a3 * (b3 x c3)

Alternatively, you can calculate the s**calar triple product** using the **determinant** of a **3×3** matrix composed of the vectors **a**, **b**, and **c**:

a · (b x c) = det( [[a1, a2, a3], [b1, b2, b3], [c1, c2, c3]] )

where **“det”** represents the **determinant** of a matrix.

The result you get from this computation is a **scalar** (a single number), which is why this operation is called the **scalar triple product**. It can be used to determine whether the vectors **a**, **b**, and **c** are **coplanar** (if the result is zero, they are coplanar) or to calculate the **parallelepiped volume** formed by these vectors.

**Exercise**

**Example 1**

Given the following vectors **a = [1, 2, 3], b = [4, 5, 6],** and **c = [7, 8, 9]**, compute **a · (b x c)**.

### Solution

First, we calculate the cross product **b x c** as given:

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

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

Then, calculate the dot product **a · (b x c)** as given:

a · (b x c) = [1, 2, 3] · [-3, 6, -3]

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

a · (b x c) = -3 + 12 – 9

a · (b x c) = 0

**Example 2**

Given the following vectors** a = [1, 0, 0], b = [0, 1, 0]**, and** c = [0, 0, 1],** calculate **a · (b x c)**.

### Solution

First, we compute the cross product **b x c** as:

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

b x c = [1, 0, 0]

Then, calculate the dot product **a · (b x c)** as:

a · (b x c) = [1, 0, 0] · [1, 0, 0]

a · (b x c) = (1 * 1) + (0 * 0) + (0 * 0)

a · (b x c)= 1

**Example 3**

Determine **a · (b x c)** for the following vectors **a = [1, 1, 1], b = [2, 3, 4], **and** c = [-1, 0, 1]**.

### Solution

First, we compute the cross product **b x c** as:

b x c = [ (3 * 1 – 4 * 0), -(2 * 1 – 4 * -1), (2 * 0 – 3 * -1) ]

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

Then, calculate the dot product **a · (b x c)** as:

a · (b x c) = [1, 1, 1] · [3, -6, 3]

a · (b x c) = (1 * 3) + (1 * -6) + (1 * 3)

a · (b x c) = 0

**Example 4**

Determine **a · (b x c)** for the following vectors **a = [5, 6, 7], b = [7, 8, 9], **and** c = [11, 12, 13]**.

### Solution

First, we compute the cross product **b x c** as:

b x c = [ (8 * 13 – 9 * 12), -(7 * 13 – 9 * 11), (7 * 12 – 8 * 11) ]

b x c = [4, 2, -4]

Then, calculate the dot product **a · (b x c)** as:

a · (b x c) = [5, 6, 7] · [4, 2, -4]

a · (b x c) = (5 * 4) + (6 * 2) + (7 * -4)

a · (b x c) = 20 + 12 – 28

a · (b x c) = 4

**Example 5**

Determine **a · (b x c)** for the following vectors **a = [1, 1, 1], b = [2, 2, 2],** and **c = [3, 3, 3]**.

**b x c** will yield [0, 0, 0] because the cross product of any two collinear vectors (vectors lying along the same line) is a zero vector.

Then, calculate the dot product **a · (b x c)** as:

a · (b x c) = [1, 1, 1] · [0, 0, 0]

a · (b x c) = (1 * 0) + (1 * 0) + (1 * 0)

a · (b x c) = 0

**Example 6**

Determine **a · (b x c)** for the following vectors **a = [2, 3, 5], b = [7, 11, 13], **and** c = [17, 19, 23]**.

### Solution

First, we compute the cross product **b x c** as:

b x c = [ (11 * 23 – 13 * 19), -(7 * 23 – 13 * 17), (7 * 19 – 11 * 17) ]

b x c = [21, 14, -7]

Then, calculate the dot product **a · (b x c)** as:

a · (b x c) = [2, 3, 5] · [21, 14, -7]

a · (b x c) = (2 * 21) + (3 * 14) + (5 * -7)

a · (b x c) = 42 + 42 – 35

a · (b x c) = 49

**Example 7**

Determine **a · (b x c)** for the following vectors **a = [1, -1, 2], b = [3, -1, 1], **and** c = [2, 1, -3]**.

### Solution

First, we compute the cross product **b x c** as:

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

b x c = [2, 7, 5]

Then, calculate the dot product **a · (b x c)** as:

a · (b x c) = [1, -1, 2] · [2, 7, 5]

a · (b x c) = (1 * 2) + (-1 * 7) + (2 * 5)

a · (b x c) = 2 – 7 + 10

a · (b x c) = 5

**Example 8**

Determine **a · (b x c)** for the following vectors **a = [4, -5, 6], b = [-7, 8, -9], **and** c = [10, -11, 12].**

### Solution

First, we compute the cross product **b x c** as:

b x c = [ (8 * 12 – -9 * -11), -(-7 * 12 – -9 * 10), (-7 * -11 – 8 * 10) ]

b x c = [1, -14, -9]

Then, calculate the dot product **a · (b x c)** as:

a · (b x c) = [4, -5, 6] · [1, -14, -9]

a · (b x c) = (4 * 1) + (-5 * -14) + (6 * -9)

a · (b x c) = 4 + 70 – 54

a · (b x c) = 20

**Applications **

Scalar triple products have wide-ranging applications in various fields. These include:

**Physics**

In **physics**, the scalar triple product is often used to calculate **volumes**, especially in problems involving physical quantities in three dimensions. For instance, it can be used to calculate the volume of a **parallelepiped** that might represent a physical system. Additionally, **electromagnetic theory** aids in understanding the properties and interactions of **electric** and **magnetic fields**.

**Engineering**

In **engineering** fields such as civil, mechanical, and electrical, the **scalar triple product** comes in handy when analyzing and solving **three-dimensional problems** related to **structure**, **fluid dynamics**, **electromagnetism**, and other areas.

**Computer Graphics**

The **scalar triple product** has a significant role in computer graphics and visualization. It is used in algorithms for rendering **3D objects**, testing the orientation of **3D models**, and detecting **collinearity** and **coplanarity** of points in a **3D space**.

**Robotics**

The **scalar triple product** in robotics is used in **motion planning**, especially in problems involving the movement of **robot arms** in **three-dimensional space**.

**Geology and Geophysics**

These fields often require **calculations** related to the **volume** of **certain geological formations** or **seismic data analysis**, where the **scalar triple product** finds application.

**Astronomy and Space Science**

The **scalar triple product** can calculate volumes and analyze spatial relations in **three-dimensional space**, which is particularly relevant in **astronomy** and **space science**.

**Mathematics**

In mathematics, the **scalar triple product** is a fundamental concept in **linear algebra**, **vector calculus**, and **differential geometry**. It can be used to solve systems of linear equations, determine whether three given vectors are coplanar, and analyze the geometrical and topological properties of mathematical objects in **3D space**.

**Machine Learning**

In **machine learning**, especially in **deep learning**, the **scalar triple product** may be used for **tensor computations**, which are vital in developing and training neural network models.

Therefore, the **scalar triple product** is a critical tool that finds utility across various disciplines. Its ability to help determine **spatial relationships** and **calculate volumes** in **three-dimensional space** is invaluable to both theoretical understanding and **practical** applications.