Vector Algebra and Foundational Geometry - Vectors (Undergraduate Foundation)

This course provides a comprehensive, first-principles guide to vector analysis. We begin with the essential definitions, distinguishing between scalars, vectors, and various vector types. You will then master the core rules of vector algebra—including the triangle and parallelogram laws of addition and scalar multiplication—before applying them to prove geometric properties of polygons. The curriculum progresses logically through key concepts such as position vectors, resolving vectors into Cartesian components, the division of lines, vector projections, and the calculation of centroids. The true power of vectors lies in their ability to connect abstract algebra to tangible geometry. This course is built around that connection, with a heavy emphasis on practical application. Through dozens of meticulously worked examples, you will move beyond theory and develop a deep, intuitive understanding of how to represent and solve complex geometric problems, building the confidence and skill needed for more advanced study. This programme is designed for students in late secondary school (such as A-Level or IB) and first-year university students in physics, engineering, mathematics, and computer science. It is the perfect starting point for anyone new to the subject and an invaluable refresher for those who need to solidify their foundational knowledge before tackling more advanced topics like linear algebra or vector calculus.

356

21 hrs

Payment required for enrolment
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.
[OAU, Ife] MTH 104: Vectors
[OAU, Ife] MTH 104: Vectors
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.

[FUTA, Akure] MTS 104: Introductory Applied Mathematics
[FUTA, Akure] MTS 104: Introductory Applied Mathematics
This learning track is designed for first-year students at the Federal University of technology, Akure (FUTA) and aligns with the second-semester coverage of introductory applied mathematics. It opens with vectors—what they are, how they work, and where they show up in real-world problems. From there, you’ll explore the geometry of circles and conic sections, gradually building up to the core ideas in basic dynamics. The lessons are short, clear, and practical—just the way we like it on UniDrills. Everything’s broken down to help you build strong intuition and problem-solving skills, especially if this is your first time engaging with applied math in this form. If you're not a FUTA student, no worries. The structure and explanations are broadly relevant, and the track works just as well for anyone looking to master these foundational topics in science and engineering.

This learning track is designed for first-year students at the Federal University of technology, Akure (FUTA) and aligns with the second-semester coverage of introductory applied mathematics. It opens with vectors—what they are, how they work, and where they show up in real-world problems. From there, you’ll explore the geometry of circles and conic sections, gradually building up to the core ideas in basic dynamics. The lessons are short, clear, and practical—just the way we like it on UniDrills. Everything’s broken down to help you build strong intuition and problem-solving skills, especially if this is your first time engaging with applied math in this form. If you're not a FUTA student, no worries. The structure and explanations are broadly relevant, and the track works just as well for anyone looking to master these foundational topics in science and engineering.

Course Chapters

1
Introduction

Definitions of scalars, vectors and tensors; representation of a vector by a directed line segment; kinds of vectors - free, localized, equal, null, unit, like and unlike vectors.

Chapter lessons

1.Welcome7:57

Welcome to the course and course outline.

2.Definition57:41

Meaning of scalars, vectors and tensors; representation of a vector by a directed line segment.

3.Kinds of vectors (1)19:29

Free and localized vectors.

4.Kinds of vectors (2)8:03

Equal and null vectors.

5.Kinds of vectors (3)10:04

Unit vectors, like and unlike vectors.

2
Vector Algebra

Vector addition - triangle and parallelogram laws; multiplication of a vector by a scalar; relations on mid-points of sides of a triangle; vector algebra on quadrilaterals and other polygons; parallel vectors; laws of vector algebra.

Chapter lessons

1.Vector addition (1)20:02

Triangle rule of addition of two vectors.

2.Vector addition (2)15:48

Parallelogram rule of addition of two vectors.

3.Vector addition (3)13:07

Polygon rule of vector addition.

4.Scalar multiplication14:06

Multiplication of a vector by a scalar.

5.Laws of vector algebra16:00

Laws (properties) of vector addition and scalar multiplication.

6.Parallel vectors27:49

Meaning and relations of parallel vectors; parallel and anti-parallel, like and unlike vectors.

7.Worked examples (1)25:55

Worked examples on vector algebra and its geometric applications.

8.Worked examples (2)29:09

More worked examples on vector algebra and its geometric applications.

9.Worked examples (3)38:21

More worked examples on vector algebra and its geometric applications.

10.Worked examples (4)35:02

More worked examples on vector algebra and its geometric applications.

11.Worked examples (5)30:16

More worked examples on vector algebra and its geometric applications.

3
Position Vectors

Meaning and algebra of position vectors.

Chapter lessons

1.Definition10:04

Meaning and representation of position vectors.

2.Worked examples (1)28:03

Worked examples on position vectors.

3.Worked examples (2)1:02:37

More worked examples on position vectors.

4
Vector Components

Meaning of vector components; resolution of vectors into components in two and three dimensions; unit vectors, direction cosines and angle between two vectors in the three-dimensional Cartesian coordinate system.

Chapter lessons

1.Definition17:56

Meaning and illustration of the components of a vector along arbitrary directions.

2.Cartesian components (1)18:48

Components of a vector in two-dimensional Cartesian coordinates.

3.Cartesian components (2)43:36

Components of a vector in three-dimensional Cartesian coordinates.

4.Direction cosines22:13

Meaning of direction cosines and the use of direction cosines to find the angle between two vectors.

5.Worked examples (1)20:08

Worked examples on vector components in two and three dimensions.

6.Worked examples (2)16:24

More worked examples on vector components in two and three dimensions.

7.Worked examples (3)15:54

More worked examples on vector components in two and three dimensions.

8.Worked examples (4)32:58

More worked examples on vector components in two and three dimensions.

9.Worked examples (5)30:40

More worked examples on vector components in two and three dimensions.

5
Division of a Line

Ratio Division of a line internally and externally; collinearity of points.

Chapter lessons

1.Internal division17:45

Internal division of a line in a given ratio by a point.

2.External division24:43

External division of a line in a given ratio by a point.

3.Collinearity21:26

Meaning of collinearity and the algebraic condition for collinearity of three points.

4.Worked examples (1)53:54

Worked examples on ratio division of a line and collinearity.

5.Worked examples (2)32:30

More worked examples on ratio division of a line and collinearity.

6.Worked examples (3)40:26

More worked examples on ratio division of a line and collinearity.

6
Vector Projections

Projection of a vector on another vector; projection of a vector on a plane.

Chapter lessons

1.On another vector20:58

Meaning of projection; analysis of the projection of a vector on another vector.

2.Onto a plane18:15

Analysis of the projection of a vector on a plane.

3.Worked examples (1)31:15

Worked examples on vector projections.

4.Worked examples (2)38:10

More worked examples on vector projections.

7
Centroids

Mean centre (geometric centre) of a number of points; weighted mean centres.

Chapter lessons

1.Centroid8:07

Meaning and analysis of the centroid of a number of points.

2.Weighted mean12:21

Meaning and analysis of the weighted mean of a number of points.

3.Worked examples (1)15:07

Worked examples on centroids and weighted means of a number of points.

4.Worked examples (2)21:15

More worked examples on centroids and weighted means of a number of points.