Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 32• If k < 0, the equation is not satisfied by any values of x, y, and z, so it has no graph.Theorem 3.1 An equation of the formx 2 +y 2 +z 2 +Gx+Hy +Iz +J = 0represents a sphere, a point, or has no graph.Cylindrical SurfacesTheorem 3.2 An equation that contains only two of the variables x, y, and z represents acylindrical surface in an xyz-coordinate system. The surface can be obtained by graphingthe equation in the coordinate plane of the two variables that appear in the equation andthen translating that graph parallel to the axis of the missing variable.Example 3.3 Sketch the graph of x 2 +z 2 = 1 in 3-space.Solution .........Example 3.4 Sketch the graph of z = siny in3-space.Solution .........3.2 VectorsMany physical quantities such as area, length, mass, and temperature are completely describedonce the magnitude of the quantity is given. Such quantities are called scalar.Other physical quantities, called vectors are not completely determined until both magnitudeand a direction are specified.Vectors can be represented geometrically by arrows in 2-space or 3-space: the directionof the arrow specifies the direction of the vector and the length of the arrow describes itsmagnitude. The tail of the arrow is called the the initial point of the vector, and the tipof the arrow the terminal point.We will denote vectors with lowercase boldface type such as a, k, v, w, and x. Whendiscussing vectors, we will refer to real numbers as scalars. Scalar will be denoted bylowercase italic type such as a, k, w, and x.Two vectors, v and w, are considered to be equal (also called equivalent) if theyhave the same length and same direction, in which case we write v = w. Geometrically,two vectors are equal if they are translations of one another; thus, the three vectors in thefollowing figure are equal, even though they are in different positions.