Scalar, Vector and Triple Products of Vectors (Undergraduate Foundation)

This course moves beyond basic vector algebra to cover the three critical methods of vector multiplication and their geometric applications. We systematically analyse the scalar (dot) product and vector (cross) product, followed by both scalar and vector triple products. The curriculum concludes by consolidating these skills to solve abstract vector equations and formulate the vector equations of straight lines in three dimensions. These operations are not abstract; they are the language of physical science and engineering. The scalar product is the standard tool for calculating work done by a force and projecting vectors. The vector product is indispensable for defining torque, angular momentum, and magnetic forces. Mastery of these products allows for precise analysis of forces, rotations, and spatial relationships in real-world systems. Upon completion, you will calculate scalar, vector, and triple products for any given vectors. You will apply these products to solve geometric problems, including testing for perpendicularity and parallelism and finding the area of a parallelogram. Critically, you will master the techniques required to solve complex vector equations and derive the vector equation of a line using various input conditions. This course is designed for first-year undergraduate students in Engineering, Physics, and Applied Mathematics. It requires prior knowledge of basic vector addition and scalar multiplication. This programme provides the necessary rigorous foundation in vector products, making it a critical prerequisite for subsequent study in classical mechanics, electromagnetism, and linear algebra.

23

15 hrs

Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
MTH 103: Elementary Mathematics III - Vectors, Geometry and Dynamics
MTH 103: Elementary Mathematics III - Vectors, Geometry and Dynamics
This comprehensive learning track guides you through the complete world of vector analysis. We begin with the fundamentals of vector algebra and its application to foundational geometry. You will then master scalar, vector, and triple products before using them to construct the vector equations of lines, planes, and conics. The journey culminates in advanced topics, including vector calculus, its applications in classical mechanics, and an introduction to differential geometry. Vectors are the essential language used to describe our physical world, making their mastery non-negotiable for any serious student of science or engineering. This track is designed to build your intuition for spatial reasoning and equip you with a powerful problem-solving toolkit. You will see direct applications in mechanics, analyzing forces and motion; in geometry, calculating angles and distances; and in calculus, modeling dynamic change over time. While this track is tailored to the first-year university curriculum for MTH 104 at Obafemi Awolowo University, Ile-Ife, Nigeria, it is an invaluable resource for a wide range of learners. It is ideal for any undergraduate student in mathematics, physics, engineering, or computer science seeking a comprehensive understanding of vector analysis. Furthermore, it serves as an excellent and thorough refresher for professionals who wish to solidify their foundational knowledge of this critical subject.

This comprehensive learning track guides you through the complete world of vector analysis. We begin with the fundamentals of vector algebra and its application to foundational geometry. You will then master scalar, vector, and triple products before using them to construct the vector equations of lines, planes, and conics. The journey culminates in advanced topics, including vector calculus, its applications in classical mechanics, and an introduction to differential geometry. Vectors are the essential language used to describe our physical world, making their mastery non-negotiable for any serious student of science or engineering. This track is designed to build your intuition for spatial reasoning and equip you with a powerful problem-solving toolkit. You will see direct applications in mechanics, analyzing forces and motion; in geometry, calculating angles and distances; and in calculus, modeling dynamic change over time. While this track is tailored to the first-year university curriculum for MTH 104 at Obafemi Awolowo University, Ile-Ife, Nigeria, it is an invaluable resource for a wide range of learners. It is ideal for any undergraduate student in mathematics, physics, engineering, or computer science seeking a comprehensive understanding of vector analysis. Furthermore, it serves as an excellent and thorough refresher for professionals who wish to solidify their foundational knowledge of this critical subject.

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Course Chapters

1. Introduction
2

Welcome and overview of vector products.

Chapter lessons

1-1. Welcome
6:04

Welcome to the course and course outline.

1-2. Overview
11:23

Meaning, types and need for vector products.

2. Scalar Products
4
5

Scalar (dot) product of two vectors and its properties.

Chapter lessons

2-1. Definition
3:34

Formal definition of the scalar or dot product of two vectors, and its relation to the projection of a vector on another vector.

2-2. Perpendicular vectors
17:12

Scalar products of perpendicular vectors and unit vectors; scalar product of two vectors in terms of their mutually-perpendicular components.

2-3. Direction cosines
24:18

Relationships between the scalar product, direction cosines of vectors and the angle between two vectors.

2-4. Properties
16:14

Properties of the scalar product of two vectors.

3. Vector Products
5
4

Vector (cross) product of two vectors and its properties.

Chapter lessons

3-1. Definition
24:07

Definition, notations and direction of the vector or cross product of two vectors.

3-2. Parallel vectors
37:00

Vector product of parallel and anti-parallel (like and unlike) vectors and unit vectors; vector product of two vectors in terms of their Cartesian components.

3-3. Collinearity
11:20

Re-examining the collinearity of two vectors (three points) in the light of cross products.

3-4. Geometric meaning
9:44

A geometric interpretation of the magnitude of a cross product of two vectors as the area of some parallelogram.

3-5. Properties
9:33

Properties of the vector product of two vectors.

4. Scalar Triple Products
4
4

Scalar triple product of three vectors and its properties.

Chapter lessons

4-1. Definition
21:19

Formal definition of the scalar triple product and its definition in terms of Cartesian components.

4-2. Geometric meaning
15:36

Geometric meaning of the scalar triple product of three vectors as the volume of some parallelepiped.

4-3. Linear dependence
29:55

Examining the linear dependence (or coplanarity) of three vectors by their scalar triple product.

4-4. Properties
12:58

Properties of the scalar triple product of three vectors.

5. Vector Triple Products
3
3

Vector triple product of three vectors and its properties.

Chapter lessons

5-1. Definition
12:31

Formal definition of the vector triple product or box product of three vectors.

5-2. Formula
40:20

Simplifying the vector triple product using scalar coefficients.

5-3. Properties
8:40

Properties of the vector triple product of three vectors.

6. Solving Vector Equations
1
8

This chapter consolidates the skills from the scalar, vector, and triple product chapters by introducing methods for solving vector equations. We analyse equations containing either unknown vectors or unknown scalars. Mastery of these techniques translates abstract vector theory into a practical problem-solving tool, which is critical for applied mechanics. By the end of this chapter, you will master: identifying the appropriate solution techniques for different equation types; solving vector equations for the unknown vector; and solving vector equations to find unknown scalar quantities.

Chapter lessons

6-1. Techniques
44:43

An overview of the solution techniques for vector equations with unknown vectors or unknown scalars.

7. Vector Equations of Straight Lines
3
4

This concluding chapter applies the vector products mastered previously to geometric analysis in three dimensions. We focus specifically on formulating and analysing the vector equations of straight lines. Mastery of this process is the crucial final step in translating abstract vector algebra into a practical tool for spatial geometry. By the end of this chapter, you will master: deriving the vector equation of a line using direction and a point; deriving the equation using two points; and determining the shortest perpendicular distance from a point to a line.

Chapter lessons

7-1. Introduction
14:07

An introduction to the vector equations of geometries.

7-2. Direction and one point
21:53

Vector equation of a straight line through a given point in a given direction.

7-3. Two points
21:23

Vector equation of a straight line through two given points.