Mathematics
Points Lines and Planes
Points, lines, and planes are fundamental geometric concepts. A point is a location in space with no size, a line is a straight path that extends infinitely in both directions, and a plane is a flat surface that extends infinitely in all directions. These concepts form the basis of geometry and are used to define and describe shapes and figures.
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7 Key excerpts on "Points Lines and Planes"
- Carl D. Crane, III, Michael Griffis, Joseph Duffy(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
1 Geometry of Points, Lines, and Planes . . . and then geometry will become what geometry ought to be. Mr. Querulous Ball’s “A Dynamical Parable” (1887) 1.1 Introduction Points, lines, and planes are the fundamental elements of spatial geometry. A point can be thought of as a location in 3D space, and its coordinates have units of length. A line can be considered to be an infinite collection of points defined by a direction (which is a dimensionless vector) that passes through some given point (which has units of meters). A plane is a two-dimensional set of points that can be defined, for example, by three points or by a line and one point. This chapter introduces the concept of homogeneous coordinates as applied to points, lines, and planes. The homogeneous coordinates of each will be defined together with the equation for each. The equation of a point, line, or plane will be shown to be a vector equation where any vector that satisfies that equation is a member of that point, line, or plane. 1.2 The Position Vector of a Point The position vector to a point Q 1 from a reference point O will be referred to as r 1 and can be expressed in the form r 1 = x 1 i + y 1 j + z 1 k w 1 (1.1) or r 1 w 1 = S O1 , (1.2) where S O1 = x 1 i + y 1 j + z 1 k and the components of the vector S O1 have units of length. The term w 1 is dimensionless. In Figure 1.1 it is assumed that w 1 = 1 and (x 1 , y 1 , z 1 ) are the usual Cartesian coordinates for the point Q 1 . The coordinates r of some general point Q may be expressed as 2 1 Geometry of Points, Lines, and Planes Figure 1.1 Coordinates of a point r w = S O , (1.3) where S O = x i + y j + z k. The subscript O has been introduced to signify that S O is origin dependent. Clearly, if we choose some other reference point, the actual point Q would not change. However, the coordinates (x , y , z), which determine Q, would change. The ratios x/w, y/w, and z/w are three independent scalars and, therefore, there are ∞ 3 points in space.
eBook - PDFCollege Geometry
A Unified Development
- David C. Kay(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
Lines, Distance, Segments, and Rays 15 (1) There are four points and four lines. (2) Each line contains exactly two points. 3. Find a geometric model for the following axioms, and determine whether or not it is a projective plane: (1) There are 5 points and 10 lines. (2) Each line contains exactly two points. (3) Each point lies on exactly four lines. 4. Find a geometric model for the following axioms, and determine whether or not it is a projective plane (see Example 2 for ideas; you can also “dualize” the geometry of Problem 3): (1) There are 10 points and 5 lines. (2) Each line contains four points. (3) Each point lies on exactly two lines. (4) If two lines intersect, they do so in only one point. 5. A well-known puzzle involves a landscape designer who wants to plant 10 fruit trees in an orchard so that there are 4 trees in each row and the outermost trees lie on a circle. How should the trees be planted? (See Problem 4.) 6. Verify that the diagram in Figure P.6 provides a geometric model for the nine-point square geometry of Example 3 by verifying each of the axioms stated in Example 3. (This diagram makes use of a famous theorem of Euclidean geometry known as Pappus’ theorem, proven later, in Chapter 7. Thus, this geometry possesses not only a geo-metrically faithful model, but one in which the lines are represented by straight line segments .) Figure P .6 7. A geometry is defined by letting the points be labeled A , B , C , …, I (the first nine letters of the alphabet) and taking as lines those sets of points whose labels occur as a horizontal row or vertical column in either of the following matrices: A B C E F D I G H A H F D B I G E C Show that this geometry is identical to one of those intro-duced in this section. 16 College Geometry: A Unified Development Group B 8. Why must the following proposition be true in any affine plane: If a line intersects one of two parallel lines, it must intersect the other also ? 9.
eBook - ePubFigures of Thought
Mathematics and Mathematical Texts
- David Reed(Author)
- 2003(Publication Date)
- Routledge(Publisher)
As has been shown, points, lines and straight lines can be completely and satisfactorily related to one another through the definitions themselves. Because of this they do not pose by themselves interesting mathematical problems for Euclid and do not constitute the subject matter of the Elements. None of this can be said for surfaces. 1.3 Surfaces Definition 5 of surfaces mimics Definition 2 of lines with two parts, length and width, instead of length alone. The remarks above on the definition of line carry over with no substantial modifications. Clearly, once again, spatial intuition is not at issue. Nor should it be thought that a specific combination of length and breadth in a notion of area is referred to. The two measurables, length and breadth, exist independently and, at least at this stage of the argument, there is no way to put them together. Definition 6 mimics Definition 3 in specifying lines as the extremities of surfaces as points are of lines. However, the grammar of the plural ‘lines’ differs from that of points. As noted above, it was precisely in functioning as the extremities of lines that any distinctions between points could be drawn. From their definition alone there are no characteristics which would permit either numerical or generic distinctions among them. But lines do have a variety of ways in which the singular/plural distinction can be applied: • lines can differ by length, • some lines can be distinguished as straight lines, • lines can act as the extremities of surfaces (allowing distinctions among lines of the same length). Now in delimiting surfaces, lines act as extremities in that they limit the double comparison of length and breadth to one term. This does not imply or involve a ‘picture’ of lines ‘cutting’ surfaces but simply a formal relating of definitions to each other
eBook - PDF- (Author)
- 2015(Publication Date)
- For Dummies(Publisher)
Top Five Elements of a Cartesian Plane ✓ Two lines cross one another at right angles to form four sections or quadrants. ✓ The two lines, or axes (pronounced ax‐eez), are number lines usually marked with the integers (positive and negative whole numbers and 0). The positives go upward on the vertical axis and to the right on the horizontal axis. ✓ The line going left and right — the horizontal line — is the x‐axis; the line going up and down — the vertical line — is the y‐axis. ✓ The little marks on the axes are called tick marks. They’re all uniformly spaced and are usually labelled with the integers, negative to positive, left to right, and downward to upward, with 0 in the middle, at the point where the axes meet. ✓ The four quadrants are numbered I, II, III and IV, with capital roman numerals starting with the upper‐right quadrant and going anticlockwise. Part IV Applying Algebra and Understanding Geometry In this part . . . ✓ Work with the Cartesian plane and understand graphing basics. ✓ Investigate intersecting lines and lines that never meet, and slide down slopes and around circles. ✓ Get the measure of perimeters, area and volume. ✓ Find out everything you need to know about geometry and those tricky triangles. Chapter 12 Graphing Basics In This Chapter a Understanding the Cartesian plane a Pointing at points and calling them by name a Graphing formulas and equations A picture is worth a thousand words. This saying is especially true in algebra. Pictures or graphs give you an instant impression of what’s happening in a situation or what an equation is representing in space. A graph is a drawing that illustrates an algebraic operation, equation or formula in a two‐dimensional plane (like a piece of graph paper). A graph allows you to see the characteristics of an algebraic statement immediately, compared to the many words needed to describe what you see in a graph.
eBook - ePub- Edward T Walsh(Author)
- 2014(Publication Date)
- Dover Publications(Publisher)
and so on.Example 1 The diagram shows several representations of lines. We will represent them in this manner throughout the book.Suppose and are two lines that intersect.3 How would you describe their intersection? Draw a picture of this situation. Your picture should help you answer the question and lead you to make the following conjecture.THEOREM 1 If two lines intersect, their intersection contains exactly one point.We can verify this conjecture on the basis of our first three postulates; but because, at this juncture, the logic involved is fairly complicated, we will postpone the task.DEFINITION 2.1 The points of a set are said to be collinear if there is a line which contains all of them. Otherwise, they are said to be noncollinear.If P ∈ , it is common to say that “P is on .” Obviously, then, A and B are on .Example 2 In the figure, A, B, and C are collinear, whereas A, B, and D are noncollinear.The next postulate introduces the undefined term plane into the discussion. Notice that this postulate and its successor are analogous to postulates P2 and P3.4POSTULATE 4 A plane is a set of points and contains at least three noncollinear points.Example 3 We will adopt the common practice of using a figure such as the one shown here to represent a plane. We generally name the plane using a lowercase letter. Thus the plane depicted would be called “plane m.”POSTULATE 5 The Plane Postulate: If P, Q, and R are three noncollinear points, then there is exactly one plane that contains all of them.It is common usage to say that sets of points “lie in” or “lie on” a plane if the plane contains them, as in the following postulate.POSTULATE 6 If two points of a line lie in a plane, then the line lies in that plane.Example 4 The situation described by P6 is depicted in the accompanying figure. Since P and Q are elements of plane m, then lies in plane m
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Unit 19 INTRODUCTION TO PLANE GEOMETRY 523 C D Figure 19–11 A B DEFINITE DISTANCE BETWEEN POINTS Figure 19–12 A B D C x x Figure 19–13 Perpendicular lines meet or intersect at a right or 90 8 angle. The symbol ' means perpendicular. Figure 19–14 shows examples of perpendicular lines. AB · ' CD · and EF ' EG . A B D C 90 8 F E G 90 8 Figure 19–14 Oblique lines are neither parallel nor perpendicular. They meet or intersect at an angle other than 90 8 , as in Figure 19–15. Figure 19–15 UNIT EXERCISE AND PROBLEM REVIEW 1. Define geometry. 2. Name the kind of surface used in plane geometry. 3. Identify the postulate that applies to each of these statements. a. If a 5 5 and b 5 5 , then a 5 b . b. If EF 5 GH , then EF 2 KL 5 GH 2 KL . c. Refer to Figure 19–16. x 5 AB 1 BC 1 CD A B D C x Figure 19–16 d. If m 5 p , then m 1 8 5 p 1 8 . e. If BC 5 DE , then 15 BC 5 15 DE . f. If e 5 AB 1 BC 1 CD , then e 2 g 5 AB 1 BC 1 CD 2 g . g. If HK 2 4 DE 5 25 , and 2 LM 1 ST 5 25 , then HK 2 4 DE 5 2 LM 1 ST . 4. Write each statement using symbols. a. Segment BC is parallel to segment DE . b. Line FG is perpendicular to line HK . c. Segment AB is parallel to line CD . Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 524 Section IV FUNDAMENTALS OF PLANE GEOMETRY 5. Sketch each of the following statements. a. Line MP is parallel to line RS , and the distance between the two lines is represented by x . b. Segment AB is perpendicular to segment CD , and point C lies on segment AB . c. Oblique lines EF and GH intersect at point R . d.
eBook - PDFA Collection of Problems in Analytical Geometry
Analytical Geometry in the Plane
- D. V. Kletenik, W. J. Langford, E. A. Maxwell(Authors)
- 2016(Publication Date)
- Pergamon(Publisher)
P A R T I Analytical Geometry in the Plane This page intentionally left blank C H A P T E R I ELEMENTARY PROBLEMS IN PLANE ANALYTICAL GEOMETRY § 1. The Axis and Intervals on an Axis. Coordinates on a Straight Line A straight line on which a positive direction has been chosen is called an axis. An interval bounded by any two points A and B on an axis is said to be directed when it is known which point is the beginning, and which point is the end, of the interval. A directed interval beginning at A and ending at B is denoted by the symbol AB. The measure of a directed inter-val on an axis is its length taken with the proper sign; the sign is plus if the direction of the interval (i.e., the direction from its beginning to its end) coincides with the positive direction of the axis, and is minus if the direction of the interval is opposite to the positive direction of the axis. The measure of the interval AB is denoted by the symbol AB, and its length by the symbol | AB |. If the points A and B coincide, the interval defined by them is called a null interval; clearly in this case AB = BA = 0 and the direction of such an interval must be considered undefined. A coordinate system can be set up on any given straight line a by: 1) the choice of a certain interval as the unit of length, 2) the designation of the positive direction on the line a, so 3 4 ANALYTICAL GEOMETRY IN THE PLANE that it becomes an axis (the positive direction on a hori-zontal axis is generally chosen to be from left to right), and 3) the choice of a particular point O as the origin of coordi-nates. The coordinate of any point M of the straight line a (in this coordinate system) is the number x which is equal to the measure of the interval OM: x = OM. The coordinate of the point O itself is 0. The symbol M(x) means that the point M has the coordinate JC.
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