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.

15 hrs

Enrolment valid for 12 months
This course is also part of the following learning tracks. You may join a 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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MTH 210: Vector Analysis
MTH 210: Vector Analysis
Vector analysis is the mathematical backbone of classical mechanics, electromagnetism, and fluid dynamics. This learning track delivers the complete NUC CCMAS MTH 210 curriculum, rigorously progressing from fundamental vector algebra to the advanced differential and integral calculus of scalar and vector fields used in complex engineering and scientific modelling. This programme is targeted at undergraduates in engineering, physics, mathematics, and computer science. It provides the essential mathematical toolkit for students entering disciplines that rely on applied mathematics and spatial analysis, and serves as a rigorous refresher for professionals needing to solidify their command of vector principles. You will master the full spectrum of vector operations including dot, cross, and triple products, and apply them to solve geometric problems and vector equations. You will acquire the skills to analyze the differential geometry of curves using the Frenet-Serret framework and apply the powerful gradient, divergence, curl, and Laplacian operators in various coordinate systems. Completion establishes the critical mathematical foundation demanded for advanced studies in continuum mechanics, electrodynamics, and theoretical physics.

Vector analysis is the mathematical backbone of classical mechanics, electromagnetism, and fluid dynamics. This learning track delivers the complete NUC CCMAS MTH 210 curriculum, rigorously progressing from fundamental vector algebra to the advanced differential and integral calculus of scalar and vector fields used in complex engineering and scientific modelling. This programme is targeted at undergraduates in engineering, physics, mathematics, and computer science. It provides the essential mathematical toolkit for students entering disciplines that rely on applied mathematics and spatial analysis, and serves as a rigorous refresher for professionals needing to solidify their command of vector principles. You will master the full spectrum of vector operations including dot, cross, and triple products, and apply them to solve geometric problems and vector equations. You will acquire the skills to analyze the differential geometry of curves using the Frenet-Serret framework and apply the powerful gradient, divergence, curl, and Laplacian operators in various coordinate systems. Completion establishes the critical mathematical foundation demanded for advanced studies in continuum mechanics, electrodynamics, and theoretical physics.

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Course Chapters

1. Introduction
2
Welcome and overview of vector products.
Concept Overviews
2 Lessons
17:27
2. Scalar Products
4
5
Scalar (dot) product of two vectors and its properties.
Concept Overviews
4 Lessons
1:01:18
Problem Walkthroughs
5 Lessons
1:43:09
3. Vector Products
5
4
Vector (cross) product of two vectors and its properties.
Concept Overviews
5 Lessons
1:31:44
Problem Walkthroughs
4 Lessons
2:12:03
4. Scalar Triple Products
4
4
Scalar triple product of three vectors and its properties.
Concept Overviews
4 Lessons
1:19:48
Problem Walkthroughs
4 Lessons
1:30:28
5. Vector Triple Products
3
3
Vector triple product of three vectors and its properties.
Concept Overviews
3 Lessons
1:01:31
Problem Walkthroughs
3 Lessons
1:25:04
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.
Concept Overviews
1 Lesson
44:43
Problem Walkthroughs
8 Lessons
6:14:49
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.
Concept Overviews
3 Lessons
57:23
Problem Walkthroughs
4 Lessons
1:09:12