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

Vector Products

Vector triple product of three vectors and its properties.

Vector Triple Products

Scalar triple product of three vectors and its properties.

Scalar Triple Products

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

Scalar Products

Introduction

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.

Solving Vector Equations

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.

Vector Equations of Straight Lines

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.

Our mission is to simplify complex concepts by teaching from first principles. We go beyond simple problem-solving to build a deep, foundational understanding of the "why" behind the "how."

Our content is meticulously designed for undergraduates navigating the rigorous curricula of top engineering and physical science programmes in Nigeria and globally.

As a core academic partner on UniDrills, we are committed to upholding the highest standards of clarity, precision, and professional excellence in education.

Max Engineering Tutorials

An academic and graduate engineer dedicated to explaining complex concepts in mathematics, computer science and engineering. I founded UniDrills to bring this vision to life, creating a world-class e-learning marketplace with a global outlook. I apply a first-principles approach to software architecture to build a professional and intuitive learning experience.

NUC CCMAS

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.

Elementary Mathematics III

Vectors, Geometry and Dynamics

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

scalar-vector-and-triple-products-of-vectors-undergraduate

Vectors

Definition of a space, linear vector space, with examples.

Linear combinations of two or more vectors and linear spans of vector spaces.

Linear Combinations and Spans

Linear dependence and independence of vectors in a vector space.

Linear Dependence and Independence

Meaning, bases and dimensions of vector spaces derived from matrices - row, column and null spaces.

Matrix Spaces

Sums and intersections of subspaces of a vector space - their basis, dimension and other properties.

Sums and Intersections

Meaning of coordinates of a vector in a vector space with respect to a given basis, and how to find the coordinates.

Coordinates

Definition and examples of linear vector subspaces.

Vector Subspaces

Basis and dimension of linear vector spaces.

Basis and Dimension

Do you want to learn how to work with abstract spaces and transformations that preserve their structure? Do you want to understand the concepts of vector subspaces, linear combinations, linear dependence, basis, dimension, coordinates, and properties of vector spaces? Do you want to master the skills of defining and manipulating linear maps, their kernels, images, matrix representations, and transition matrices?

If you answered yes to any of these questions, then this course is for you!

Linear Algebra: Linear Vector Spaces and Linear Maps is a comprehensive and engaging course that covers the fundamentals of vector spaces and linear maps and their applications in mathematics and science. You will learn how to:

- Define and classify vector spaces and their subspaces over a given scalar field

- Perform operations on vectors using linear combinations and scalar multiplication

- Determine whether a set of vectors is linearly dependent or independent and find a basis and dimension for a vector space or subspace

- Find the coordinates of a vector with respect to a given basis and change the basis using transition matrices

- Define and classify linear maps between vector spaces and find their domains, codomains, ranges, and null spaces

- Find the kernel and image of a linear map and use them to determine whether a linear map is one-to-one or onto

- Represent a linear map using a matrix and perform matrix operations such as addition, multiplication, and inversion

- Use different methods and tools to solve systems of linear equations, such as Gaussian elimination, row reduction, and inverse matrices

This course is suitable for anyone who wants to learn or review the basics of vector spaces and linear maps and their applications. It is especially useful for students and professionals in algebra, geometry, analysis, differential equations, optimization, cryptography, computer graphics, data science, and other related fields.

By the end of this course, you will have a solid foundation of the theory and practice of vector spaces and linear maps and their operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve vector spaces and linear maps.

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

OAU, Ife

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors.

Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Mathematical Methods II

Master the algebraic structures that underpin modern science and computation. This academic track delivers the complete NUC CCMAS MTH 204 curriculum, moving rigorously from abstract vector spaces to practical matrix theory. It provides the essential mathematical toolkit required for advanced problem-solving in high-demand STEM fields.

This programme is targeted at undergraduates in mathematics, engineering, and computer science requiring a firm grounding in linear structures. It also serves professionals in data science, cryptography, and machine learning needing a rigorous theoretical refresher on foundational concepts.

You will achieve competence in manipulating abstract vector spaces, determining basis and dimension, and analyzing linear transformations through their kernels and images. You will master matrix arithmetic, compute determinants, solve systems of linear equations using advanced methods, and apply techniques of eigenvalues and diagonalization. Completion establishes the critical foundation demanded for advanced studies in multivariate calculus, differential equations, and complex computational algorithms.

Linear Algebra I

linear-vector-spaces-linear-algebra-undergraduate

Linear Algebra

Analysis of equilibrium of rigid bodies in three dimensions.

3D Equilibrium Equations

General analysis of equilibrium of rigid bodies in two dimensions.

2D Equilibrium Equations (1)

Determining force and moment reactions at contacts, connections and supports for a rigid body under the action of co-planar forces - in order to produce its free-body diagram.

2D Reactions

Determining force and moment reactions at contacts, connections and supports for a rigid body under the action of three-dimensional forces - in order to produce its free-body diagram.

3D Reactions

Different conditions of constraints on rigid bodies in two dimensions and their statical determinacy.

2D Constraints and Statical Determinacy

Welcome to the course and review of fundamental concepts - internal and external forces, equilibrium conditions for rigid bodies, use of free-body diagrams, etc.

Different conditions of constraints on rigid bodies in three dimensions and their statical determinacy.

3D Constraints and Statical Determinacy

Analysis of special cases of equilibrium of rigid bodies in two dimensions - equilibrium under the action of only two or three forces.

2D Equilibrium Equations (2)

Equilibrium is the foundation of all structural and mechanical analysis. This course provides a rigorous study of the equilibrium of rigid bodies in two and three dimensions. It covers reactions, equilibrium equations, constraints, and statical determinacy, supported by extensive worked examples.

Engineers, architects, and applied scientists constantly rely on equilibrium principles to design safe, efficient, and reliable systems. From bridges and buildings to machines and vehicles, every stable structure depends on correct modelling of supports, forces, and constraints. Mastery of equilibrium ensures accuracy in analysis and prevents costly or dangerous failures.

By completing this course, you will be able to draw free-body diagrams, apply equilibrium equations in 2D and 3D, and classify and analyse different support and constraint conditions. You will understand when a system is properly constrained, over-constrained, or indeterminate, and you will have practical experience solving a wide range of equilibrium problems.

This course is intended for undergraduate students in mechanical, civil, and structural engineering, but it is also useful to anyone studying applied mechanics at an advanced level. It provides both a theoretical foundation and practical problem-solving skills valuable to practising engineers, technical instructors, or anyone needing precise understanding of equilibrium in rigid body mechanics.

Master the science of structural stability. This programme provides a complete education in engineering statics, covering the analysis of forces, moments, and equilibrium in rigid bodies. It delivers the non-negotiable principles required to design safe and reliable structures.

This learning track is for first or second-year undergraduate students in Civil, Mechanical, Aerospace, and Structural Engineering. It is the foundational mechanics course upon which all subsequent design and analysis subjects are built.

Analyse the forces within any static structure, from simple trusses to complex machines. You will master free-body diagrams, equilibrium equations, and structural analysis methods. This programme equips you with the core analytical skills for a career in structural, mechanical, or civil design.

Engineering Mechanics

Statics

Master the non-negotiable principles of Engineering Mechanics. This track delivers a rigorous, complete programme in statics and dynamics, built to the NUC GET 207 core curriculum. It moves methodically from force systems and equilibrium to the kinematics of rigid bodies and the kinetics of particles. This is the foundation of all structural and mechanical analysis.

This programme is for first and second-year undergraduate engineering students. It is an essential requirement for students in Mechanical, Civil, Structural, Aerospace, and Mechatronics engineering. A working knowledge of introductory physics and calculus is assumed.

On completion, you will be able to analyse and solve complex problems in statics and particle dynamics. You will draw free-body diagrams, apply equilibrium equations, analyse trusses and frames, and solve motion problems using Newton's laws, work-energy, and momentum methods. This programme prepares you for advanced courses, particularly Mechanics of Materials, and future professional engineering practice.

Applied Mechanics

equilibrium-of-rigid-bodies-engineering-mechanics

Kinematics of curvilinear motion of a point on a rigid body undergoing rotation about a fixed axis, and its applications in gears, pulleys, etc.

Rotation About a Fixed Axis (2)

Kinematics of motion of a rigid body undergoing general plane motion using the velocity of a point relative to another point on the same rigid body.

Relative Velocity Analysis

Kinematics of angular motion of a rigid body (or a line on a rigid body) undergoing rotation about a fixed axis.

Rotation About a Fixed Axis (1)

Acceleration analysis of the motion of a rigid body undergoing general plane motion using the acceleration of a point relative to another point on the same rigid body.

Relative Acceleration Analysis

Analysis of motion of a rigid body undergoing general plane motion by locating an instantaneous centre of zero velocity.

Instantaneous Centre of Rotation

Meaning of rigid bodies; different kinds of motion undergone by rigid bodies.

Kinematics of general plane motion of rigid bodies by relating the angular motion of a line to the motion of a point on the same rigid body or different rigid bodies with dependent motion.

Absolute Motion Analysis

Kinematics of motion of a rigid body undergoing general plane motion using parameters measured relative to a reference frame in rotation.

Motion Relative to a Rotating Frame

In this course, you will learn the fundamentals of two-dimensional kinematics of rigid bodies, which is the branch of mechanics that deals with the geometry of the motion of rigid bodies without considering the forces that cause it. You will learn how to use different coordinate systems and reference frames to express the position, velocity, and acceleration of various points on a rigid body in plane motion.

Empower yourself with the knowledge and skills needed to excel in diverse fields such as mechanical engineering, robotics, aerospace engineering, and beyond. Imagine designing a robotic arm that moves with precision, or optimizing the performance of a rotating machinery. These are some of the countless applications of rigid-body kinematics you will learn in this course.

By the end of this course, you will be able to:

- Understand the rotation of rigid bodies about fixed axes and its applications

- Analyze absolute motion to understand how rigid bodies move through space

- Explore relative velocity analysis to solve complex motion problems

- Master the concept of instantaneous centres of rotation for precise motion analysis

- Understand relative acceleration analysis to predict the behavior of moving bodies

- Analyze motion relative to a rotating frame to unravel the dynamics of rotating systems

Whether you're a seasoned professional looking to expand your knowledge or a curious beginner eager to explore the motion of rigid bodies, this course is for you.

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

Master the principles governing the motion of engineering systems. This learning track provides a complete education in dynamics, systematically building from the kinematics of particles to the kinetics of rigid bodies and culminating in an introduction to dynamic systems and vibration. You will learn to analyse and predict how mechanical systems behave under the influence of forces.

This programme is for undergraduate students in mechanical, aerospace, civil, or related engineering disciplines. It is also essential for practising engineers and applied scientists who require a rigorous, first-principles command of dynamic analysis for their professional work. A prerequisite knowledge of statics, calculus, and vector algebra is assumed.

Upon completion, you will possess the analytical tools to solve complex dynamics problems for particles and rigid bodies using force, energy, and momentum methods. This provides the essential foundation for advanced study in mechanical design, control systems, and structural analysis, and prepares you for demanding technical roles in the engineering sector.

Dynamics

2-d-kinematics-of-rigid-bodies-engineering-mechanics

This chapter teaches the fundamental grammar of HTML. We define the core vocabulary of elements and tags, construct a valid page structure, and add basic, semantic content. These are the non-negotiable building blocks for any web page.

Key topics: defining HTML vocabulary (tags, elements); building a valid page structure; adding content with semantic text; and understanding element display behaviour.

The Language of HTML

This chapter establishes a professional workflow for asset management. We will source, create, and organise all required media - logos, headshots, and icons - before building the page. A structured and legally compliant asset base is essential.

Key topics: understanding asset copyright; structuring the project folder; creating a logo and headshot; and choosing a modern icon library.

Preparing Your Project Assets

This final chapter covers the essential "behind-the-scenes" tags in the document <head> that make your page professional, complete, and ready for the world. These details control how your site appears in search engines and social media, and prepare it for styling.

By the end, you will: understand the role of the <head> element; add essential meta tags for SEO and social sharing; set a custom favicon for your site; and understand how to link a CSS file for the next course.

Finalising the Document

This chapter focuses on the correct, professional way to display structured data. We will build a complete "Experience" table for our portfolio, including an interactive, expandable section for details.

By the end, you will: build a complete, semantically correct table; and use the details and summary elements to add interactive, expandable content to your table rows.

Displaying Tabular Data

This chapter focuses on building the high-level skeleton of our portfolio page using semantic HTML. We will translate the main visual sections of our design into the correct, professional HTML5 structural elements.

By the end, you will: structure your page with header and footer landmarks; create a navigation bar with the nav element; and use main and section elements to define the primary content areas of your page.

Building the Page Structure

With the main structure in place, this chapter is about populating our sections with their actual content. We will add the text, images, and icons that make up the hero, skills, and social media sections of our portfolio.

By the end, you will: build a complete hero section with text and an image; understand the critical attributes of the img element; implement SVG icons; and make links functional.

Adding Core Content and Media

This chapter covers the essential HTML for collecting user input. We will build the complete structure for the portfolio's contact form, a non-negotiable component for enabling professional communication. You will learn to handle everything from simple text fields to complex option selectors, ensuring your form is both functional and accessible.

By the end, you will: build a complete and accessible form; use various input types for different data, including text, email, and options; semantically group related inputs using fieldset and legend; and provide a functional submission button.

Building the Contact Form

This chapter focuses on a critical professional skill: building repeatable components. We will build the HTML structure for the "Project Card" from our design, integrating different media types for a dynamic result.

By the end, you will: understand the role of the div element; build a reusable project card structure; and embed static images, local videos, and third-party videos into your project.

Structuring Complex Components

This chapter establishes your professional environment. We will set up the project with version control, generate a design target, and configure a live-preview workflow. This is the standard setup required before writing any code.

Key topics: initialising a project folder and Git repository; generating a target UI Design; and configuring an instant-preview workflow with Live Server.

HTML is not about memorizing tags; it is the professional practice of structuring information. This course teaches HTML5 by guiding you through the construction of a complete, portfolio-ready webpage. We move from a blank file, through a professional asset-preparation phase, to building a multi-section, feature-rich website based on a modern UI design.

A well-structured portfolio is a non-negotiable tool for any aspiring developer. The quality of your HTML directly impacts your site's performance, search ranking, and accessibility. This course teaches you to write the clean, semantic, and professional markup that separates a hobbyist's page from an impressive portfolio piece that gets you noticed.

You will master the full lifecycle of a modern HTML project. You will prepare all necessary assets using AI and design tools. You will structure a complete page with semantic landmarks, build complex components like project cards and interactive tables, and create a functional contact form. By the end, you will have a complete, well-structured project.

This course is designed for the absolute beginner with no prior coding experience. It is the essential starting point for aspiring developers and is invaluable for designers, content creators, and marketers who need to build or understand the fundamental structure of a professional website.

Beginner

This is the direct path to building professional-grade web development skills. This learning track is a comprehensive programme covering the entire modern front-end stack, engineered to take you from zero knowledge to possessing a practical, portfolio-worthy skillset. It moves logically from foundational web concepts and professional tools, through structuring content with HTML5, styling with modern CSS, and finally, building dynamic user experiences with JavaScript.

This programme is built for individuals starting a career in technology with no prior coding experience. It is the essential curriculum for aspiring front-end developers, full-stack engineers, and computer science students. It also serves professionals in adjacent roles - such as UI/UX designers and product managers - who require a practical understanding of web technologies to excel in their careers.

By the end of this learning track, you will have the ability to build, style, and launch a fully interactive, responsive website from a blank file. You will master the professional developer's toolkit - Git, the command line, and an AI-assisted editor - and write clean, semantic code. This track provides the foundational project and technical skills to confidently begin your career path toward junior developer roles and to build robust personal or business projects.

Modern Web Development Foundations

beginner-modern-web-development-foundations

From Zero to Your First Interactive Website

html-foundations-from-zero-to-your-first-web-page

Web Development

This chapter sets the stage for styling our portfolio. We will connect our stylesheet to the project and establish the core concepts and workflow you will use to bring your design to life.

By the end, you will: understand the role of CSS; connect a stylesheet to your project; and be prepared to translate your visual design into code.

This chapter covers the fundamental 'physics' of CSS. Using simple, isolated examples, you will master the core concepts of selectors, the box model, typography, colours, and the rules of specificity before we apply them to our main project.

By the end, you will: master the most common CSS selectors; define and use colours effectively; control typography; understand the box model; and explain how CSS specificity works.

CSS Fundamentals

This chapter covers the critical initial setup for any professional web project. You'll establish a consistent styling foundation, ensuring your design appears predictably across all browsers, and centralize core design values for efficient project management.

By the end of the chapter, you will master CSS resets, define global design tokens, and integrate custom web fonts to precisely match your design's aesthetic.

Setting Global Styles

This final chapter provides a comprehensive review of the project and a clear roadmap for your continued learning. We will recap the key CSS concepts applied to build the portfolio page and look ahead to the next steps in your journey as a web developer.

Key topics include a review of the responsive layouts and styled components you created. We will then outline a path for further practice, introduce advanced CSS concepts like Grid and animations, and explain how JavaScript will bring your project to life.

Conclusion

This chapter introduces Flexbox, the essential tool for modern layouts. We will use it to arrange and align the components of our portfolio page, transforming our vertical stack of elements into a professional layout that matches our design.

By the end, you will: understand the core concepts of Flexbox; align items horizontally and vertically; and build the main layouts for our portfolio's header, hero, and skills sections.

Layout with Flexbox

With the main page layout complete, this chapter focuses on styling individual components. We will apply the visual polish required to transform our raw structure into a professional user interface, moving from broad architecture to detailed aesthetics.

Key topics include styling cards with shadows and spacing, making tables readable with custom borders and colours, creating consistent form elements, and adding interactive hover effects with transitions.

Styling the Components

This chapter covers the position property, a tool for taking elements out of the normal document flow. While Flexbox manages the main layout, mastering positioning is essential for creating complex, layered user interfaces like sticky headers and overlay menus.

Key topics include using the position property's values (sticky, absolute, relative), creating positioning contexts to anchor elements, and managing overlapping layers with the z-index property to control stacking order.

Element Context

Plain HTML is functional, but unstyled pages do not engage users. This course provides comprehensive training in modern CSS, guiding you from fundamental syntax and the box model to advanced, responsive layouts. You will transform a raw document into a professionally styled, multi-component webpage from scratch, learning the complete workflow for styling a real-world project.

Mastery of CSS is a non-negotiable skill for any web professional. The techniques covered are directly applicable to building professional portfolios, business websites, or the front end of any web application. This course equips you with the practical skills to translate design concepts into responsive, production-ready code for any device, ensuring your projects look polished and professional.

You will learn to write clean CSS and master selectors, specificity, and the box model. The course covers building a complete mobile-first layout with Flexbox and creating a consistent design system with CSS variables. You will finish by styling individual components like cards and forms, and controlling element context with the `position` property and `z-index`.

This programme is built for anyone with a basic knowledge of HTML, including aspiring front-end developers and university students. Experienced programmers will find it a direct path to modernising their layout skills, while UI/UX designers will gain a practical understanding of the medium for which they design.

css-foundations-style-your-first-web-page

Force-acceleration analysis of rectilinear motion of a particle.

Rectilinear Motion (1)

Meaning of mechanics, dynamics and kinetics, overview of Newton's second law and impulse-and-momentum methods; units of measurement of mass and force.

Force-acceleration analysis of curvilinear motion of particles using radial and transverse components.

Curvilinear Motion (3)

Force-acceleration analysis of curvilinear motion of particles using normal and tangential components.

Curvilinear Motion (2)

Force-acceleration analysis of curvilinear motion of particles in rectangular coordinates.

Curvilinear Motion (1)

Force-acceleration analysis of absolute and relative motion of particles connected by rigid links or light, inextensible strings passed over pulleys.

Rectilinear Motion (3)

Force-acceleration analysis of absolute and relative rectilinear motion of two or more particles in direct contact.

Rectilinear Motion (2)

This course will equip you with the skills and knowledge to tackle various engineering and scientific problems involving kinetics of particles, which is the branch of mechanics that deals with the relation between the forces acting on a particle and its resulting motion. From the basics of mechanics, dynamics, and kinetics to getting comfortable with Newton's second law and momentum methods, you will clearly understand how forces cause motion.

Boost your problem-solving skills and open up new career opportunities in fields like aerospace engineering, mechanical engineering, civil engineering, biomedical engineering, physics, chemistry, and more. Imagine being able to predict how a ball flies through the air, or understanding why a car behaves the way it does in a crash. This course helps you do these and many more!

By the end of this course, you will be able to:

- Explain the fundamentals of mechanics, dynamics, and kinetics

- Use Newton's second law to analyze how forces make bodies speed up or slow down

- Learn about energy, how forces do work, and how these relate to motion of bodies

- Figure out how collisions work and what motion happens before and after collisions

This course is designed for students, engineers, scientists, and anyone who is interested in learning the basics of kinetics of particles. The course assumes that you have a basic background in calculus, physics, vector algebra, and statics.

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

kinetics-of-particles-by-newtons-second-law-engineering

Introduction to and motivation for a dedicated study of the kinetics of systems of particles.

Generalized Newton's second law application to kinetics of systems of particles.

Newton's Second Law

Analysis of motion in a system of particles involving steady mass flow into or out of it.

Steady Streams of Particles

Analysis of motion in a system of particles gaining or losing mass.

Systems Gaining or Losing Mass

Work-Energy principle and conservation of energy, impulse-momentum principle and conservation of momentum, for a system of particles.

Energy and Momentum Methods

Analysis of kinetics of generalized systems of particles by Newton's second law, work-energy principle and impulse-and-momentum methods.

kinetics-of-systems-of-particles-engineering-mechanics

Meaning of the range of a linear map, with worked examples.

Range

Meaning of the kernel of a linear map, with worked examples.

Kernel

Definition, examples and general properties of linear maps.

Linear Maps

Matrix representation of a linear map, with worked examples.

Matrix Representation

Meaning and properties of the transition matrix between two bases of a vector space.

Transition Matrix

Definition of a map (also mapping or transformation) between two sets, notations and some general properties of maps.

Maps

Further operations on linear maps involving their definition and properties.

Operations on Linear Maps

Analysis of the vector space of all linear maps between two given vector spaces. Addition, scalar multiplication, composition and inverse of linear maps.

Algebra of Linear Maps

Welcome and course outline. Review of elementary matrix and linear vector space concepts.

A comprehensive study of linear transformations (or maps) - definition, kernel, range, matrix representation and related matters.

linear-maps-transformations-linear-algebra

Meaning, operations and properties of eigenvalues and eigenvectors of matrices.

Eigenvalues and Eigenvectors

Diagonalization of matrices; evaluating polynomials and transcendentals of matrices.

Diagonalization of Matrices

Quadratic and canonical forms; transformations using symmetric matrices and orthogonal diagonalizing matrices.

Quadratic and Canonical Forms

Welcome and course outline. Review of elementary matrix concepts.

Diagonalization of symmetric matrices - orthogonal diagonalizing matrix, the Gram-Schmidt orthogonalization procedure, and related concepts.

Symmetric Matrices

This course provides a clear and structured exploration of eigenvalues, eigenvectors, and diagonalization, focusing on both theory and real-world applications. 

You???ll learn how to compute eigenvalues and eigenvectors, understand their geometric significance, and apply diagonalization to simplify complex matrix operations. 

Topics include linear transformations, dynamical systems, and applications in physics, engineering, and machine learning. 

The course is designed for students, engineers, and data scientists seeking a strong foundation in matrix methods. By the end, you'll confidently apply these concepts in problem-solving and computational modeling.

Advanced engineering mathematics covering solid analytical geometry, integrals, scalar and vector fields, matrices and determinants and complex variables.

Curated for third-year students of engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Engineering Analysis I

Advanced linear algebra is the mathematical backbone of modern data science, engineering, and physics. This learning track delivers the rigorous MTH 205 curriculum based on NUC CCMAS standards, focusing on sophisticated matrix analysis and practical computational methods critical for solving complex technical problems.

This programme is targeted at undergraduates in mathematics, engineering, and computer science requiring a deep command of advanced matrix theory. It is equally essential for data scientists and engineers seeking a rigorous theoretical foundation for machine learning algorithms, cryptography, and complex system modelling.

You will master matrix manipulations to solve linear systems and compute determinants and inverses efficiently using various methods including software like Python and MATLAB. You will gain competence in determining eigenvalues and eigenvectors, applying diagonalization to analyze the stability of dynamical systems, and working with quadratic and canonical forms. Completion establishes the critical mathematical expertise required for advanced studies in multivariate statistics, differential equations, and algorithmic development.

Linear Algebra II

eigenvalues-eigenvectors-and-diagonalization-of-matrices

Courses

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

Linear Vector Spaces - Linear Algebra (Undergraduate Advanced)

Equilibrium of Rigid Bodies - Engineering Mechanics (Undergraduate Advanced)

2D Kinematics of Rigid Bodies - Engineering Mechanics (Undergraduate Advanced)

HTML Foundations: From Zero To Your First Web Page - Web Development (Professional Practice)

CSS Foundations: Style Your First Web Page - Web Development (Professional Practice)

Kinetics of Particles by Newton's Second Law - Engineering Mechanics (Undergraduate Advanced)

Kinetics of Systems of Particles - Engineering Mechanics (Undergraduate Advanced)

Linear Maps (Transformations) - Linear Algebra (Undergraduate Advanced)

Eigenvalues, Eigenvectors and Diagonalization of Matrices - Linear Algebra (Undergraduate Advanced)