Vector Algebra and Geometry - Introduction to Vectors

Do you want to learn how to work with quantities that have both magnitude and direction? Do you want to understand the concepts of scalars, vectors, tensors, components, projections, products, and equations of vectors? Do you want to master the skills of performing algebraic, geometric, differential, and integral operations on vectors using different methods and tools? If you answered yes to any of these questions, then this course is for you! This course covers the fundamentals of vectors and their applications in mathematics and mechanics. You will learn how to: - Define and classify scalars, vectors, and tensors and their properties - Represent vectors by directed line segments, unit vectors, direction cosines, and coordinates - Perform vector addition, subtraction, and multiplication by a scalar using the triangle and parallelogram laws - Apply vector algebra to various geometrical problems involving mid-points, parallelism, and collinearity - Find the position vector of a point and use it to locate the point in space - Resolve vectors into components in two and three dimensions and use them to simplify vector operations - Divide a line in a given ratio internally or externally using vectors - Project a vector on another vector or a plane and use it to find the angle or distance between them - Find the centroid of a set of points or a polygon using vectors - Find the scalar and vector products of two vectors and use them to calculate the area, volume, and orthogonality of geometrical figures - Find the scalar and vector triple products of three vectors and use them to determine the coplanarity and linear dependence of vectors - Solve vector equations with unknown vectors or scalars using various techniques - Find the vector equation of a line or a plane and use it to describe the direction, intersection, angle, and distance of lines and planes - Find the parametric and non-parametric equations of circles, parabolas, ellipses, and hyperbolas using vectors - Differentiate and integrate vector-valued functions and use the rules of vector differentiation and integration - Apply vector differentiation and integration to find the derivatives, arc length, and curvature of parametric curves - Find the tangential, normal, and binormal vectors and the osculating, normal, and rectifying planes of a parametric curve using the Frenet-Serret formulas - Apply vectors to various problems in mechanics, such as forces, equilibrium, work, energy, momentum, displacement, velocity, acceleration, and motion in different coordinate systems and frames of reference This course is suitable for anyone who wants to learn or review the basics of vectors and their applications. It is especially useful for students and professionals in engineering, physics, geometry, calculus, and other related fields. By the end of this course, you will have a firm understanding of vectors and their operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve vectors. 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.

315

21 hrs

$ 7.15

Payment required for enrolment
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 104: Vectors
MTH 104: Vectors
Introduction to vectors, covering vector algebra, geometry, products, vector equations of geometries, vector differentiation, integration and their applications to mechanics and differential geometry. Curated for first-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.

Introduction to vectors, covering vector algebra, geometry, products, vector equations of geometries, vector differentiation, integration and their applications to mechanics and differential geometry. Curated for first-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.

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.