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	Time: 20 hours Level: Intermediate

	Introduction Introduction Resource 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.

	1 Coordinate geometry: points, planes and lines 1.1 Points, lines and distances in two-dimensional Euclidean space Resource In coordinate geometry we generally use rectangular (or Cartesian) coordinate axes, as illustrated below, to describe the Euclidean plane. We can represent any point in the plane uniquely by an ordered... 1.2 Lines Resource The equation of any line in 2, except a line parallel to the y-axis, can be written in the form 1.3 Parallel and perpendicular lines Resource We often wish to know whether two lines are parallel (that is, they never meet) or perpendicular (that is, they meet at right angles). 1.4 Intersection of two lines Resource Two arbitrary lines in 2 may have a single point of intersection, may be parallel, or may coincide. The first two possibilities are illustrated below. Can we tell from the equations of the lines which... 1.5 Distance between two points in the plane Resource Next, we find the formula for the distance between two points P (x1, y1) and Q(x2, y2) in the plane. In the diagram below we have drawn P and Q in the first quadrant, but the formula we derive holds wherever... 1.6 Points, planes, lines and distances in three-dimensional Euclidean space Resource We now study three-dimensional space, 3. This is a space with which you are familiar, of course, as ‘the real world’ is a three-dimensional space. 1.7 Planes in three-dimensional Euclidean space Resource We now look at the general form of the equation of a plane in 3. 1.8 Intersection of two planes Resource We saw earlier that two arbitrary lines in 2 may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in 3 may intersect, be parallel, or coincide. 1.9 Distance between points in three-dimensional Euclidean space Resource You saw in Section 1.5 that the distance between two points (x1, y1) and (x2, y2) in the plane is given by 1.10 Further exercises Resource Determine the equation of the line through each of the following pairs of points. Show that both equations can be written in the form ax + by = c, for some real numbers a, b and c, where a and b are not...

	2 Vectors 2.1 Definitions Resource In this section we introduce an alternative way of describing points in the plane 2 or in three-dimensional space 3; namely, by using vectors. 2.2 Multiplication by a scalar Resource In the collection of vectors sketched in Section 2.1, although v is not equal to c, the vectors v and c are closely related: c is a vector in the same direction as v, but it is twice as long as v. Thus... 2.3 Addition of vectors Resource We saw above that the vector 2v can be regarded as the vector v ‘followed by’ the vector v; we can also quite naturally describe this vector as being the ‘sum’, v + v, of the vector with itself. 2.4 Components and the arithmetic of vectors Resource We introduce now a different method of representing vectors, which will make the manipulation of vectors much easier. Thus we shall avoid having to solve problems involving vectors by drawing the vectors... 2.5 Position vectors Resource Finally, we relate the method of specifying points in the plane as an ordered pair of real numbers (that is, via the Cartesian coordinate system) to vectors. 2.6 Lines Resource Earlier, we found the equation of a line in the (x, y)-plane in the form ax + by = c, for some real numbers a, b and c, where a and b are not both zero. We now find an equivalent equation for a line... 2.7 Further exercises Resource Let p = 2i − 3j + k and q = −i −2j −4k be two vectors in 3. Determine p + q, p − q and 2p − 3q.

	3 Dot product 3.1 Definition, properties and some applications Resource In the previous section we saw how to add two vectors and how to multiply a vector by a scalar, but we did not consider how to multiply two vectors. There are two different ways in which we can multiply... 3.2 Post-audio exercises Resource Let u and v be the position vectors (6, 8) and (−12, 5), respectively. 3.3 Equation of a plane in three-dimensional Euclidean space Resource We stated in Section 1.7 that the general form of the equation of a plane in 3 is 3.4 Further exercises Resource (a) Find the angle between each of the pairs of vectors:

	4 Conics 4.1 Conic sections Resource Conic section is the collective name given to the shapes that we obtain by taking different plane slices through a double cone. The shapes that we obtain from these cross-sections are drawn below. It is... 4.2 Circles Resource Recall that a circle in 2 is the set of points (x, y) that lie at a fixed distance, called the radius, from a fixed point, called the centre of the circle. We can use the techniques of coordinate geometry... 4.3 Focus–directrix definitions of the non-degenerate conics Resource Earlier, we defined the conic sections as the curves of intersection of planes with a double cone. One of these conic sections, the circle, can be defined as the set of points a fixed distance from a fixed... 4.4 Parabola (e = 1) Resource A parabola is defined to be the set of points P in the plane whose distances from a fixed point F are equal to their distances from a fixed line d. We obtain a parabola in standard form if 4.5 Ellipse (0 Resource An ellipse with eccentricity e (where 0 4.6 Hyperbola (e > 1) Resource A hyperbola is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d, where e > 1. We obtain a hyperbola in standard form if 4.7 Rectangular hyperbola (e = √2) Resource If the eccentricity e of a hyperbola is equal to √2, then e2 = 2 and b = a. Then the asymptotes of the hyperbola have equations y = ±x, so they are at right angles. A hyperbola whose asymptotes are at... 4.8 General equation of a conic Resource You have already met the parabola, ellipse and hyperbola. So far, you have considered the equation of a conic only when it is in standard form; that is, when the centre of the conic (if it has a centre)... 4.9 Further exercises Resource Determine the equation of the circle with centre (2, 1) and radius 3.

	References and Acknowledgements Acknowledgements Resource

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