Vector Equations of Lines, Planes, and Conics

This course explores a crucial application of vector analysis: describing and solving problems in geometry. We begin by mastering the techniques for solving abstract vector equations before applying these skills to define geometric shapes. You will learn to construct the vector equations for lines, planes, and the conic sections (circles, ellipses, parabolas, and hyperbolas) in both two and three dimensions. The ability to describe complex geometries with concise equations is a cornerstone of modern science and engineering. This course bridges the gap between abstract vector theory and its practical application in modeling the real world. By working through a vast library of examples, you will learn to analyze the relationships between shapes—calculating intersections, angles, and distances—with precision and confidence. This advanced course is designed for students ready to apply their knowledge of vector products to analytical geometry. A thorough understanding of vector algebra, dot products, and cross products is essential. This programme is the critical bridge to higher-level topics, making it ideal for university students preparing for courses in vector calculus or mechanics.

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.

[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.

Course Chapters

Introduction

Welcome and course outline.

Chapter lessons

1.Welcome10:41

Welcome to the course and outline of course.

Solving Vector Equations

Solutions of vector equations with unknown vectors; solutions of vector equations with unknown scalars.

Chapter lessons

1.Techniques44:43

An overview of the solution techniques for vector equations with unknown vectors or unknown scalars.

2.Worked examples (1)49:33

Worked examples on vector equations with unknown vectors.

3.Worked examples (2)55:19

More worked examples on vector equations with unknown vectors.

4.Worked examples (3)37:43

More worked examples on vector equations with unknown vectors.

5.Worked examples (4)51:02

More worked examples on vector equations with unknown vectors.

6.Worked examples (5)26:05

More worked examples on vector equations with unknown vectors.

7.Worked examples (6)44:26

More worked examples on vector equations with unknown vectors.

8.Worked examples (7)56:22

Worked examples on vector equations with unknown scalars.

9.Worked examples (8)54:19

More worked examples on vector equations with unknown scalars.

Straight Lines

Direction vector and vector equation of a straight line,; angle between two straight lines; intersecting lines, parallel lines and skew lines; shortest distance between two skew lines.

Chapter lessons

1.Introduction14:07

An introduction to the vector equations of geometries.

2.Direction and one point21:53

Vector equation of a straight line through a given point in a given direction.

3.Two points21:23

Vector equation of a straight line through two given points.

4.Worked examples (1)12:00

Worked examples on the vector equation of a straight line.

5.Worked examples (2)10:47

More worked examples on the vector equation of a straight line.

6.Worked examples (3)13:42

More worked examples on the vector equation of a straight line.

7.Worked examples (4)32:43

More worked examples on the vector equation of a straight line - how to obtain the shortest (perpendicular) distance from a point to a given straight line.

8.Angle between two lines

How to find the angle between two lines from their vector equations.

9.Intersection

Intersection of two lines; intersection of a line with a plane.

10.Skew lines

Meaning of skew lines and the perpendicular distance between two skew lines.

11.Worked examples (5)

More worked examples on the vector equation of a straight line.

12.Worked examples (6)

More worked examples on the vector equation of a straight line.

13.Worked examples (7)

More worked examples on the vector equation of a straight line.

14.Worked examples (8)

More worked examples on the vector equation of a straight line.

15.Worked examples (9)

More worked examples on the vector equation of a straight line.

Planes

Vector equations of a plane; normal vector of a plane; parallel planes and the distance between them; angle between two planes; angle between a line and a plane.

Chapter lessons

1.A point and two lines

Vector equation of a plane through a given point and parallel to two given lines.

2.Three non-collinear points

Vector equation of a plane containing three given non-collinear points.

3.A point and a normal vector

Vector equation of a plane containing a given point and normal to a given vector (or line).

4.Worked examples (1)

Worked examples on vector equations of a plane.

5.Worked examples (2)

More worked examples on vector equations of a plane.

6.Worked examples (3)

More worked examples on vector equations of a plane.

7.Parallel planes

Parallel planes, their equations and minimum (perpendicular) distance apart.

8.Angle between two planes

How to find the angle between two intersecting planes.

9.Intersection with a line

Re-examining the point of intersection of a line and a plane; angle between a line and a plane.

10.Intersection of two planes

How to find the equation of the line of intersection of two planes.

11.Worked examples (4)

More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.

12.Worked examples (5)

More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.

13.Worked examples (6)

More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.

14.Worked examples (7)

More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.

15.Worked examples (8)

More worked examples on the vector equation of a plane, intersections and angles between a line and plane, and between two planes.

Circles

Vector equations of a circle in x-y plane, a circle in space; non-parametric vector and standard forms; equation of the plane containing a circle.

Chapter lessons

1.x-y plane

Vector equation of a circle in x-y plane.

2.Three dimensions

Vector equation of a circle in a three-dimensional space.

3.Non-parametric equation

General non-parametric of equation of a circle in a plane; non-parametric of equation of a circle in the x-y plane.

4.Worked examples (1)

Worked examples on vector equations of a circle.

5.Worked examples (2)

More worked examples on vector equations of a circle.

6.Worked examples (3)

More worked examples on vector equations of a circle.

Ellipses

Parametric and non-parametric vector equations of an ellipse.

Chapter lessons

1.x-y plane

Parametric equation of an ellipse in the x-y plane.

2.Three dimensions

Parametric equation of an ellipse in three dimensions.

3.Non-parametric equations

Non-parametric equations of an ellipse in two and three dimensions.

4.Worked examples (1)

Worked examples on parametric and non-parametric equations of an ellipse.

5.Worked examples (2)

More worked examples on parametric and non-parametric equations of an ellipse.

Parabolas

Parametric and non-parametric vector equations of a parabola.

Chapter lessons

1.x-y plane

Parametric vector equations of a parabola in the x-y plane.

2.Three dimensions

Parametric vector equations of a parabola in 3 dimensions.

3.Non-parametric equations

Non-parametric equations of a parabola in two and three dimensions.

4.Worked examples (1)

Worked examples on parametric and non-parametric equations of a parabola.

5.Worked examples (2)

More worked examples on parametric and non-parametric equations of a parabola.

Hyperbolas

Parametric and non-parametric vector equations of a hyperbola.

Chapter lessons

1.x-y plane

Parametric vector equations of a hyperbola in the x-y plane.

2.Three dimensions

Parametric equation of a hyperbola in three dimensions.

3.Non-parametric equations

Non-parametric equations of a hyperbola in two and three dimensions.

4.Worked examples (1)

Worked examples on the parametric and non-parametric equations of a hyperbola.

5.Worked examples (2)

More worked examples on the parametric and non-parametric equations of a hyperbola.