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

This course delves into the essential topic of vector multiplication. We systematically explore the four main types of vector products, beginning with the scalar (dot) and vector (cross) products of two vectors. You will then advance to products involving three vectors: the scalar triple product and the vector triple product, uncovering their unique properties and applications. Vector products are the key to unlocking the geometric power of vector analysis. They provide the tools to calculate angles, projections, areas, and volumes with elegant algebraic formulas. This course focuses on revealing these deep connections, showing how abstract multiplication rules translate into powerful methods for solving real-world problems in physics and engineering. This is an intermediate course designed for students who have a solid grasp of foundational vector algebra and components. It is the perfect next step for students in A-Level or university STEM programmes looking to deepen their understanding of vector analysis. A strong foundation in the topics from our introductory course is highly recommended for success.

14

15 hrs

Payment required for enrolment
Enrolment valid for 12 months

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.