Mathematics Foundations/13.1 Points, Lines, and Planes

13.1 Points, Lines, and Planes Introduction Euclidean geometry is named after the ancient Greek mathematician Euclid, whose work Elements (c. 300 BCE) laid the foundation for geometric study for over two millennia. This section explores the fundamental building blocks of Euclidean geometry: points, lines, and planes, along with their properties and relationships. Fundamental Objects Points A point represents a location in space. It has no size, length, width, or depth—only position. Points are typically denoted by capital letters such as A, B, or P. A point is a primitive notion in Euclidean geometry Points have zero dimensions Points can be located using coordinates in a coordinate system Lines A line is a straight one-dimensional figure that extends infinitely in both directions. A line can be determined by two distinct points. A line has one dimension (length) Lines are typically denoted by lowercase letters like l, m, or by naming two points on the line, such as A line segment is a portion of a line with two endpoints, denoted as A ray is a portion of a line with one endpoint, extending infinitely in one direction, denoted as Planes A plane is a flat, two-dimensional surface that extends infinitely in all directions. A plane can be determined by three non-collinear points or by a line and a point not on the line. A plane has two dimensions (length and width) Planes are typically denoted by Greek letters like α (alpha), β (beta), or by capital letters like P A plane contains infinitely many points and lines Axioms of Euclidean Geometry Euclidean geometry is built upon a set of axioms or postulates. Here are some fundamental axioms related to points, lines, and planes: Two distinct points determine exactly one line. A line contains infinitely many points. Three non-collinear points determine exactly one plane. If two points lie in a plane, then the entire line containing those points lies in the plane. If two planes intersect, their intersection is a line. Relationships Between Geometric Objects Incidence Relations A point is incident with a line if the point lies on the line. A line is incident with a plane if the line lies in the plane. A point is incident with a plane if the point lies in the plane. Parallel and Perpendicular Lines Two lines in a plane are either: Intersecting at exactly one point, or Parallel (denoted ), never intersecting Two lines that intersect at right angles (90°) are called perpendicular (denoted ). Parallel and Perpendicular Planes Two planes are parallel if they never intersect. Two planes are perpendicular if they intersect at a right angle. A line and a plane are perpendicular if the line forms right angles with all lines in the plane that pass through the point of intersection. Distance and Angle Measures Distance Between Points The distance between two points A(x1, y1) and B(x2, y2) in a two-dimensional plane is given by: In three-dimensional space, for points A(x1, y1, z1) and B(x2, y2, z2): Angles An angle is formed by two rays sharing a common endpoint (vertex). Angles are typically measured in degrees (°) or radians (rad). A right angle measures 90° or radians An acute angle measures less than 90° An obtuse angle measures more than 90° but less than 180° A straight angle measures 180° or π radians Coordinate Systems Cartesian Coordinates In a Cartesian coordinate system: A point in a plane is represented by an ordered pair (x, y) A point in space is represented by an ordered triple (x, y, z) A line can be represented by a linear equation: (in 2D) A plane can be represented by a linear equation: (in 3D) Parametric Representations A line passing through point P0(x0, y0, z0) with direction vector can be represented parametrically as: where t is a real parameter. Applications The concepts of points, lines, and planes form the foundation for: Computer graphics and modeling Architecture and engineering design Navigation and GPS systems Physics (particularly mechanics) Computer vision and robotics Exercises 1. Prove that the distance from a point to a line in a plane is the length of the perpendicular from the point to the line. 2. Given three points A(1, 2), B(4, 6), and C(7, 2), determine whether they are collinear. 3. Find the equation of a plane passing through the points P(1, 0, 2), Q(0, 1, 3), and R(2, 1, 1). 4. Determine the angle between the lines and . 5. Prove that if two lines are parallel to the same line, then they are parallel to each other.