线性代数教学资料-cha课件.ppt

Li Jie12 Vectors in 2-space and 3-spaceLi Jie2Overview In this chapter we review the related concepts of physical vectors,geometric vectors,and algebraic vectors.To provide maximum geometric insight,we concentrate on vectors in two-space and three-space.Later,in Chapter 3,we will generalize many of the ideas developed in this chapter and apply them to a study of vectors in n-space,that is,to vectors in Rn.A major emphasis in Chapter 3 is on certain fundamental ideas such as subspaces of Rn and the dimension of a subspace.As we will see in Chapter 3,concepts such as subspace and dimension are directly related to the geometrically familiar notions of lines and planes in three-space.Li Jie3Core sectionsVectors in the planeVectors in spaceThe dot product and the cross productLines and planes in space Li Jie42.1 Vectors in the plane1.Three types of vectors(1)Physical vectors:A physical quantity having both magnitude and direction is called a vector.Typical physical vectors are forces,displacements,velocities,accelerations.Li Jie5(2)Geometric vectors:The directed line segment from point A to point B is called a geometric vector and is denoted byFor a given geometric vector ,the endpoint A is called the initial point and B is the terminal point.Li Jie6(3)Equality of geometric vectorsAll geometric vectors having the same direction and magnitude will be regarded as equal,regardless of whether or not they have the same endpoints.xyEFABCDLi Jie7(4)Position vectorsxyABOPLi Jie8(5)Components of a vectorLi Jie9Theorem2.1.1:Let and be geometric vectors.Then if and only if their components are equal.(6)An equality test for Geometric VectorsLi Jie10(7)Algebraic vectors:Theorem2.1.2:Let be a geometric vector,with A=(a1,a2)and B=(b1,b2).Then can be represented by the algebraic vector Li Jie112.Using algebraic vectors to calculate the sum of geometric vectorsTheorem2.1.2:Let u and v be geometric vectors with algebraic representations given byThen the sum u+v has the following algebraic representation:Li Jie123.Scalar multiplicationTheorem2.1.3:Let u be a geometric vectors with algebraic representations given byThen the scalar multiple cu has the following algebraic representation:Li Jie134.Subtracting geometric vectors5.Parallel vectorsVectors u and v are parallel if there is a nonzero scalar c such that v=cu.If c0,we say u and v have the same direction but if c0,we say u and v have the opposite direction.6.Lengths of vectors and unit vectors7.The basic vectors i and j2.1 Exercise P126 26Li Jie142.2 Vectors in space1.Coordinate axes in three space2.The right-hand rule3.Rectangular coordinates for points in three space axis;coordinate planes;octants4.The distance formulaTheorem2.2.1:Let P=(x1,y1,z1)and Q=(x2,y2,z2)be two points in three space.The distance between P and Q,denoted by d(P,Q),is given byLi Jie155.The midpoint formulaTheorem2.2.2:Let P=(x1,y1,z1)and Q=(x2,y2,z2)be two points in three space.Let M denote the midpoint of the line segment joining P and Q.Then,M is given by6.Geometric vectors and their components7.Addition and scalar multiplication for vectors8.Parallel vectors,lengths of vectors,and unit vectors9.The basic unit vectors in three spaceLi Jie162.3 The dot product and the cross product1.The dot product of two vectorsDefinition 2.3.2:Let u and v are two-dimensional vectors,then the dot product of u and v,denoted uv,is defined by u v=u1v1+u2v2.Let u and v are three-dimensional vectors,then the dot product of u and v,denoted uv,is defined by u v=u1v1+u2v2+u3v3.Definition 2.3.1:Let u and v are vectors,then the dot product of u and v,denoted uv,is defined by u v=|u|v|cos.where is the angle of vectors u and v.Li Jie17Li Jie182.The angle between two vectorsu v=|u|v|cos.3.Algebraic properties of the dot productLi Jie194.Orthogonal Vectors(正交向量正交向量）When=/2 we say that u and v are perpendicular or orthogonal.Theorem 2.3.1:Let u and v are vectors,then u and v are orthogonal if and only if u v=0.In the plane,the basic unit vectors i and j are orthogonal.In three space,the basic unit vectors i,j and k are mutually orthogonal.Li Jie205.Projections6.The cross productDefinition 2.3.3:Let u and v are vectors,then the cross product of u and v,denoted uv,is a vector that it is orthogonal to u and v,and u,v,uv is right-hand system,and the norm of the vector is|uv|=|u|v|sin.where is the angle of vectors u and v.qvuUnit vector,directionLi Jie217.Remember the form of the cross product(two methods)determinantLi Jie228.Algebraic properties of the cross product9.Geometric properties of the cross productLi Jie2310.Triple products（三重积）（三重积）11.Tests for collinearity and coplanarity2.3 Exercise P148 48Theorem:Let u,v and w be nonzero three dimensional vectors.(a)u and v are collinear if and only if uv=0.(b)u,v and w are coplanar if and only if u(vw).Li Jie242.4 Lines and planes in space1.The equation of a line in xy-planeyxOP0=(x0,y0).lLi Jie252.The equation of a line in three space(1)Point and directional vector form equation of a lineLi Jie26(2)Parametric equations of a lineLi Jie27Example1:Let L be the through P0=(2,1,6),having direction vector u given by u=4,-1,3T.(a)Find parametric equations for the line L.(b)Does the line L intersect the xy-plane?If so,what are the coordinates of the point of intersection?Example2:Find parametric equations for the line L passing through P0=(2,5,7)and the point P1=(4,9,8).Li Jie282.The equation of a plane in three spacePoint and normal vector form equation of a planeLi Jie29Example3:Find the equation of the plane containing the point P0=(1,3,-2)and having normal n=5,-2,2T.Example4:Find the equation of the plane passing through the points P0=(1,3,2),P1=(2,0,-1),and P2=(4,5,1).Li Jie30The relationship between two lines or two planesTwo linesA line and a planeTwo plane