4.2 Circles 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 to find the equation of a circle with a given centre and radius.
4.1 Conic sections 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 thought that the Greek mathematician Menaechmus discovered the conic sections around 350 bc. 3.4 Further exercises (a) Find the angle between each of the pairs of vectors: (3, 1) and (1, −2); i + 2j and −3i + j − 2k 3.3 Equation of a plane in three-dimensional Euclidean space We stated in Section 1.7 that the general form of the equation of a plane in 3.2 Post-audio exercises Let u and v be the position vectors (6, 8) and (−12, 5), respectively. (a) Sketch u and v on a single diagram. On th 3.1 Definition, properties and some applications 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 two vectors, known as the dot product (or scalar product) and the vector product. They are given these names because the result of the first is a scalar and the result of the second is a vector. (We shall not consider vector products in this course.) In the audio sec 2.7 Further exercises Let p = 2i − 3j + k and q = −i −2j −4k be two vectors in 2.4 Components and the arithmetic of vectors 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 and making measurements, which is very time-consuming and never very accurate. We can think of a vector as a translation, that is, as representing a movement by a certain amount in a given direction. Then we can use the Cartesian axes in the plane or in Author(s): 2.3 Addition of vectors 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. Analogously, if p is the vector 2 cm E and q is the vector 3 cm NE, we can think of the ‘sum’ p + q of the vectors as follows. Starting from a given point, O say, draw the vector p; starting from its finishi 2.2 Multiplication by a scalar 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 it is natural to write 2v for c, since we can think of a journey repre 2.1 Definitions In this section we introduce an alternative way of describing points in the plane 1.10 Further exercises 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 1.9 Distance between points in three-dimensional Euclidean space You saw in Section 1.5 that the distance between two points (x1, y1) and (x2, y2) in the plane is given by 1.8 Intersection of two planes We saw earlier that two arbitrary lines in 1.7 Planes in three-dimensional Euclidean space We now look at the general form of the equation of a plane in Three planes whose equations are easy to find are those that contain a pair of axes. For example, the (x, y)-plane is the plane that contains the x-axis and the y-axis. Points which lie in this plane are prec 1.5 Distance between two points in the plane 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 the points are in the plane.
1.4 Intersection of two lines Two arbitrary lines in
1.3 Parallel and perpendicular lines We often wish to know whether two lines are parallel (that is, they never meet) or perpendicular (that is, they meet at right angles). Two distinct lines, y = m1x + c1 and y = m2x + c2, are parallel if and only if they have the same gradient; that is, if and only if m1 = m2. For example, the lines y = −2x + 7 and 1.2 Lines The equation of any line in Learning outcomes By the end of this unit you should be able to: Section 1 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.
Exercise 47
3 is
Exercise 42
Exercise 31
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2 or in three-dimensional space 3;
Exercise 12
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2 may intersect, be parallel, or coincide. In an analogous way, two arbitrary planes in
3.
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 of the three possibilities occurs?Author(s):
2, except a line parallel to the y-axis, can be written in the form
