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    Vector Geometry
    Vector Geometry
    Vector Geometry
    Ebook323 pages2 hours

    Vector Geometry

    By Gilbert De B. Robinson

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    About this ebook

    This brief undergraduate-level text by a prominent Cambridge-educated mathematician explores the relationship between algebra and geometry. An elementary course in plane geometry is the sole requirement for Gilbert de B. Robinson's text, which is the result of several years of teaching and learning the most effective methods from discussions with students.
    Topics include lines and planes, determinants and linear equations, matrices, groups and linear transformations, and vectors and vector spaces. Additional subjects range from conics and quadrics to homogeneous coordinates and projective geometry, geometry on the sphere, and reduction of real matrices to diagonal form. Exercises appear throughout the text, with complete answers at the end.
    LanguageEnglish
    PublisherDover Publications
    Release dateOct 10, 2013
    ISBN9780486321042
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      Book preview

      Vector Geometry - Gilbert De B. Robinson

      GEOMETRY

      1

      LINES AND PLANES

      1.1 COORDINATE GEOMETRY

      The study of geometry is essentially the study of relations which are suggested by the world in which we live. Of course our environment suggests many relations, physical, chemical and psychological, but those which concern us here have to do with relative positions in space and with distances. We shall begin with Euclidean geometry, which is based on Pythagoras’ theorem:

      The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

      The statement of this fundamental result implies a knowledge of length and area as well as the notion of a right angle. If we know what we mean by length and may assume its invariance under what we call motion, we can construct a right angle using a ruler and compass. We define the area of a rectangle as the product of its length and breadth. To be rigorous in these things is not desirable at this stage, but later on we shall consider a proper set of axioms for geometry.

      While the Greeks did not explicitly introduce coordinates, it is hard to believe that they did not envisage their usefulness. The utilization of coordinates was the great contribution of Descartes in 1637, and to us now it is a most natural procedure. Take an arbitrary point O in space, the corner of the room, for instance, and three mutually perpendicular coordinate axes. These lines could be the three lines of intersection of the walls and the floor at O; the planes so defined we call the coordinate planes. In order to describe the position of a point X, we measure its perpendicular distances from each of these three planes, denoting the distances x1, x2, x3 as in Figure 1.1 It is important to distinguish direction in making these measurements. Any point within the room has all its coordinates (x1, x2, x3) positive; measurements on the opposite side of any coordinate plane would be negative. Thus the following eight combinations of sign describe the eight octants of space about O:

      FIG. 1.1

      We may describe the points on the floor by saying that x3 = 0; this is the equation of this coordinate plane. Limiting our attention to such points, we have plane geometry. If we call the number of mutually perpendicular coordinate axes the dimension of a space, then a plane has two dimensions and the position of each point is given by two coordinates, while space as we have been describing it has three dimensions.

      1.2 EQUATIONS OF A LINE

      If we assume that a line is determined uniquely by any two of its points, it is natural to seek characterizing properties dependent on these two points only. To this end we refer to Figure 1.2, assuming X to have any position on the line ZY, and complete the rectangular parallelepiped as indicated. If the coordinates of the points in question are

      and if XA, AB, AC are parallel to the coordinate axes with XD parallel to ZP, then from similar triangles,

      It follows from this proportionality that if we set ZX = τZY, then

      so that, in terms of coordinates,

      These equations may be rewritten thus:

      in which form they define the coordinates of X as linear functions of the parameter τ. Clearly, if τ = 0 then X = Z, and if τ = 1 then X = Y.

      If we set

      then l1, l2, l3 are called the direction numbers of the line l. If X and X′ are any two distinct points on l, then

      so that numbers proportional to l1, l2, l3 are determined by any two distinct points on l. Two lines whose direction numbers are proportional are said to be parallel. We can summarize these results by writing

      It follows that we may write the equations of l in the symmetric form

      or

      but it should be emphasized that these are valid only if all the denominators are different from zero, i.e., provided the line l is not parallel to one of the coordinate planes. As will appear in the sequel, it is the parametric equations 1.22 which are most significant. Moreover, they generalize easily and provide the important link between classical geometry and modern algebra.

      FIG. 1.2

      Let us now assume that ZY makes angles θ1, θ2, θ3 with ZQ, ZR, ZS, i.e., with Ox1, Ox2, Ox3. One must be careful here to insist on the direction being from Z to Y; otherwise the angles θi might be confused with π – θi. With such a convention,

      and λ1, λ2, λ3 are called the direction cosines of the line l. By Pythagoras’ theorem, ZP2 = ZQ2 + ZR2, so that

      Thus, given l1, l2, l3, we have

      and parallel lines make equal angles with the coordinate axes.

      Clearly, λ1, λ2, λ3 may be substituted for l1, l2, l3 in 1.232, and we may write the first set of equations of 1.22 in the form

      EXERCISES

      1. Find the equations, in parametric and symmetric form, of the line joining the two points Y(1, –2, –1) and Z(2, –1, 0).

      Solution. The parametric equations of the line in question are, by 1.22,

      and in the symmetric form 1.231,

      2. What are the direction cosines of the line in Exercise 1? Write the equations of the line in the form 1.27.

      3. Find parametric equations for the line through the point (1, 0, 0) parallel to the line joining the origin to the point (0, 1, 2). Could these equations be written in the form 1.232?

      4. Find the equations of the edges of the cube whose vertices are the eight points (±1, ±1, ±1), as in Figure 5 of Chapter 4.

      5. Find the direction cosines of the edges of the regular tetrahedron with vertices

      1.3 VECTOR ADDITION

      The notion of a vector in three dimensions, or 3-space, can be introduced in two ways:

      (i) A vector is a directed line segment of fixed length.

      (ii) A vector X is an ordered* triple of three numbers (x1, x2, x3), called the components of X.

      It is important to have both definitions clearly in mind. If we write a small arrow above the symbols to indicate direction, then this is determined for V = in Figure 2 by the components

      also, the length of ZY or the magnitude of V is given by

      The position of a vector is immaterial, so we may assume it to have one end tied to the origin. Sometimes a vector is called free if it can take up any position, but this distinction is not made in either (i) or (ii). In this sense a vector is more general than any particular directed segment, and could be described as an equivalence class† of directed segments.

      That the two definitions (i) and (ii) are equivalent follows from the theorem:

      1.32 Two vectors are equal if and only if their components are equal.

      Proof. Since the components (υ1, υ2, υ3) determine the magnitude and direction of a vector V, the condition is certainly sufficient. Conversely, if |U| = |V| then

      and if U and V have the same direction, we must have

      so that k2 = 1. It follows that k = 1 and the two vectors must coincide.

      Following this line of thought, we denote the vector with components (ku1, ku2, ku3) by kU so that

      k may be any real number, and |k| is k taken positive. In particular, k may be zero, in which case kU is the zero vector 0 with components (0, 0, 0). Evidently the magnitude of 0 is zero and its direction is undefined.

      We define the sum

      of two vectors U and V to be the diagonal of the parallelogram formed by U and V. Alternatively, we may define W by means of the formulas

      It will be sufficient to consider these definitions in the plane where a vector is defined by two components only. We take the vectors U(u1, u2) and V(υ1, υ2) and complete the parallelogram, as in Figure 1.3; it follows immediately that the components of W satisfy the relation 1.33. But there is more to be learned from the figure. For example, we arrive at the same result whether we go one way around the parallelogram or the other way around, so that

      FIG. 1.3

      and vector addition is commutative. This is also a consequence of the commutativity of addition as applied to the components in 1.33. Finally, by reversing the direction of U we obtain the vector –U so that

      where 0 is the zero vector. The other diagonal of the parallelogram is the vector –U + V, as indicated.

      Consider now the similarity between the formulas 1.33 defining vector addition and the parametric equations of a line in 1.27. If we denote by Z with components (z1, z2, z3) and by Λ the vector with components (λ1, λ2, λ3), then the relations 1.27 are the scalar equations equivalent to the vector equation

      It follows that the notion of a vector is of central significance in Euclidean geometry. As the title of this book suggests, our purpose is to develop these ideas in several different contexts. Some of these contexts are officially algebraic while others are geometric, but with this thread to guide us, we shall see their interrelations and why it is that mathematics is a living subject, changing and progressing with the introduction of new ideas.

      FIG. 1.4

      EXERCISES

      where O is the origin, A is the point (2, –3, 1), P is the point (4, –6, 2), Q is the point (–7, 3, 1), and R is the point (–5, 0, 2).

      where A is the point (1, 2, 3), B is the point (–2, 3, 1), and C is the point (3, –2, –4), and show that

      in Exercise 2. What would be the components of a parallel vector of unit length?

      4. If U, V, W are three arbitrary vectors, show that

      (the associative law of addition).

      5. Prove that the medians of any triangle ABC are concurrent.

      Solution. If D is the midpoint of BC. Since the centroid G divides AD in the ratio 2:1,

      Since this result is symmetric in A, B, C, the medians must be concurrent in G.

      1.4 THE INNER PRODUCT

      In the preceding section we defined the multiplication of a vector U by a scalar k. Such multiplication is called scalar multiplication and it is obviously commutative,

      There is another kind of multiplication of vectors which is of great importance. To define it we use the generalized Pythagorean theorem to yield

      so that

      FIG. 1.5

      Substituting from 1.31 and simplifying, we have

      where cos φ1, cos φ2, cos φ3 are the direction cosines of XY, and cos ψ1, cos ψ2, cos ψ3 are those of XZ. Since it is important to have a convenient expression for the sum of products appearing in 1.44, we define the inner or scalar product of the vectors U, V to be

      where U has components (u1, u2, u3) and V as components (υ1, υ2, υ3).

      All these formulas are valid also in the plane, but in this case a vector U has only two components (u1, u2), and the angles φ1, φ2 between U and the coordinate axes are complementary. Thus cos φ2 = sin φ1, so that

      and it is convenient to write the equation of a line (note that there is now only one equation),

      in the form

      Rather than try to visualize a space of more than three dimensions, one should think of a vector V as having n components (υ1, υ2, …, υn). The sum of two vectors, W = U +

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