Vectors and conics

Introduction

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This unit is an adapted extract from the course Pure mathematics (M208) [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]

The idea of vectors and conics may be new to you. In this unit 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.

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