Chapter 1 Topics in Analytic Geometry
Chapter 1 Topics in Analytic Geometry 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.
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 33If the initial point of v is A and the terminal point is B, then we write v = −→ AB whenwe want to emphasize the initial and terminal points.•BA •If the initial and terminal points of a vector coincide, then the vector has length zero;we call this the zero vector and denote it by 0. The zero vector does not have a specificdirection, so we will agree that it can be assigned any convenient direction in a specificproblem.Definition 3.1 If v and w are vectors, then the sum v+w is the vector from the initialpoint of v to the terminal point of w when the vectors are positioned so the initial point ofw is at the terminal point of v.wvv+wDefinition 3.2 If v is a nonzero vector and k is a nonzero real number (a scalar), thenthe scalar multiple kv is defined to be the vector whose length is |k| times the length ofv and whose direction is the same as that of v if k > 0 and opposite to that of v if k < 0.We define kv = 0 if k = 0 or v = 0.v2v1v 2(−1)v(− 3 2 )vObserve that if k and v are nonzero, then the vectors v and kv lie on the same line iftheir initial points coincide and lies on parallel or coincident lines if they do not. Thus, wesay that v and kv are parallel vectors. Observe also that the vector (−1)v has the samelength as v but is oppositely directed. We call (−1)v the negative of v and denote it by−v. In particular, −0 = (−1)0 = 0.Vector subtraction is defined in terms of addition and scalar multiplication byv−w = v+(−w)In the special case where v = w, their difference is 0; that is,v+(−v) = v−v = 0
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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 33If the <strong>in</strong>itial po<strong>in</strong>t of v is A and the term<strong>in</strong>al po<strong>in</strong>t is B, then we write v = −→ AB whenwe want to emphasize the <strong>in</strong>itial and term<strong>in</strong>al po<strong>in</strong>ts.•BA •If the <strong>in</strong>itial and term<strong>in</strong>al po<strong>in</strong>ts of a vector co<strong>in</strong>cide, then the vector has length zero;we call this the zero vector and denote it by 0. The zero vector does not have a specificdirection, so we will agree that it can be assigned any convenient direction <strong>in</strong> a specificproblem.Def<strong>in</strong>ition 3.1 If v and w are vectors, then the sum v+w is the vector from the <strong>in</strong>itialpo<strong>in</strong>t of v to the term<strong>in</strong>al po<strong>in</strong>t of w when the vectors are positioned so the <strong>in</strong>itial po<strong>in</strong>t ofw is at the term<strong>in</strong>al po<strong>in</strong>t of v.wvv+wDef<strong>in</strong>ition 3.2 If v is a nonzero vector and k is a nonzero real number (a scalar), thenthe scalar multiple kv is def<strong>in</strong>ed to be the vector whose length is |k| times the length ofv and whose direction is the same as that of v if k > 0 and opposite to that of v if k < 0.We def<strong>in</strong>e kv = 0 if k = 0 or v = 0.v2v1v 2(−1)v(− 3 2 )vObserve that if k and v are nonzero, then the vectors v and kv lie on the same l<strong>in</strong>e iftheir <strong>in</strong>itial po<strong>in</strong>ts co<strong>in</strong>cide and lies on parallel or co<strong>in</strong>cident l<strong>in</strong>es if they do not. Thus, wesay that v and kv are parallel vectors. Observe also that the vector (−1)v has the samelength as v but is oppositely directed. We call (−1)v the negative of v and denote it by−v. In particular, −0 = (−1)0 = 0.Vector subtraction is def<strong>in</strong>ed <strong>in</strong> terms of addition and scalar multiplication byv−w = v+(−w)In the special case where v = w, their difference is 0; that is,v+(−v) = v−v = 0