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Attempts to answer problems in areas as diverse as science, technology and economics involve solving simultaneous linear equations. In this free course, Vectors and conics, we look at some of the equations that represent points, lines and planes in mathematics. We explore concepts such as Euclidean space, vectors, dot products and conics.
After studying this course, you should be able to:
- recognise the equation of a line in the plane
- determine the point of intersection of two lines in the plane, if it exists
- recognise the one-one correspondence between the set of points in three-dimensional space and the set of ordered triples of real numbers
- recognise the equation of a plane in three dimensions
- explain what are meant by a vector, a scalar multiple of a vector, and the sum and difference of two vectors.
- Current section: Introduction
- Learning outcomes
- 1 Coordinate geometry: points, planes and lines
- 1.1 Points, lines and distances in two-dimensional Euclidean space
- 1.2 Lines
- 1.3 Parallel and perpendicular lines
- 1.4 Intersection of two lines
- 1.5 Distance between two points in the plane
- 1.6 Points, planes, lines and distances in three-dimensional Euclidean space
- 1.7 Planes in three-dimensional Euclidean space
- 1.8 Intersection of two planes
- 1.9 Distance between points in three-dimensional Euclidean space
- 1.10 Further exercises
- 2 Vectors
- 3 Dot product
- 4 Conics
- Keep on learning
Study this free course
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Vectors and conics
The idea of vectors and conics may be new to you. In this course we look at some of the ways that we represent points, lines and planes in mathematics.
In Section 1 we revise coordinate geometry in two-dimensional Euclidean space,2, and then extend these ideas to three-dimensional Euclidean space, 3. We discuss the equation of a plane in 3, but find that we do not have the tools to determine the equation of a plane, and leave this until Section 3.
In Section 2 we introduce the idea of a vector, and look at the algebra of vectors. Vectors give us a way of looking at points and lines, in the plane and in 3, which is sometimes more useful than Cartesian coordinates, although the two are closely related.
In Section 3 we introduce the idea of the dot product of two vectors, and then use it to determine the general form of the equation of a plane in 3.
In Section 4 we explain the origin of conics, as the curves of intersection of double cones and planes in 3. The focus–directrix definitions of the non-degenerate conics, the ellipse, the parabola and the hyperbola, are given. We observe that conics are precisely the subsets of the plane determined by an equation of degree two.
This OpenLearn course is an adapted extract from an Open University course
Copyright & revisions
Originally published: Wednesday, 18th May 2011
Last updated on: Tuesday, 15th March 2016
- Creative-Commons: The Open University is proud to release this free course under a Creative Commons licence. However, any third-party materials featured within it are used with permission and are not ours to give away. These materials are not subject to the Creative Commons licence. See terms and conditions. Full details can be found in the Acknowledgements and our FAQs section.
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