线性代数教学资料-cha课件

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1、2 Vectors in 2-space and 3-space,Overview,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 id

2、eas 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 di

3、rectly related to the geometrically familiar notions of lines and planes in three-space.,Core sections,Vectors in the plane Vectors in space The dot product and the cross product Lines and planes in space,2.1 Vectors in the plane,1. Three types of vectors,Physical vectors: A physical quantity having

4、 both magnitude and direction is called a vector. Typical physical vectors are forces, displacements, velocities, accelerations.,(2) Geometric vectors: The directed line segment from point A to point B is called a geometric vector and is denoted by,For a given geometric vector ,the endpoint A is cal

5、led the initial point and B is the terminal point.,(3) Equality of geometric vectors,All geometric vectors having the same direction and magnitude will be regarded as equal, regardless of whether or not they have the same endpoints.,(4) Position vectors,(5) Components of a vector,Theorem2.1.1: Let a

6、nd be geometric vectors. Then if and only if their components are equal.,(6) An equality test for Geometric Vectors,(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,2. Using algebraic vectors to calculate th

7、e sum of geometric vectors,Theorem2.1.2: Let u and v be geometric vectors with algebraic representations given by,Then the sum u+v has the following algebraic representation:,3. Scalar multiplication,Theorem2.1.3: Let u be a geometric vectors with algebraic representations given by,Then the scalar m

8、ultiple cu has the following algebraic representation:,4. Subtracting geometric vectors,5. Parallel vectors Vectors 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 v

9、ectors and unit vectors,7. The basic vectors i and j,2.1 Exercise P126 26,2.2 Vectors in space,1. Coordinate axes in three space,2. The right-hand rule,3. Rectangular coordinates for points in three space axis;coordinate planes;octants,4. The distance formula,Theorem2.2.1: Let P=(x1,y1,z1) and Q=(x2

10、,y2,z2) be two points in three space. The distance between P and Q, denoted by d(P,Q), is given by,5. The midpoint formula,Theorem2.2.2: 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 by,6. Geometric vec

11、tors and their components,7. Addition and scalar multiplication for vectors,8. Parallel vectors, lengths of vectors, and unit vectors,9. The basic unit vectors in three space,2.3 The dot product and the cross product,1. The dot product of two vectors,Definition 2.3.2: Let u and v are two-dimensional

12、 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 de

13、fined by u v=|u| |v| cos. where is the angle of vectors u and v.,2. The angle between two vectors,u v=|u| |v| cos.,3. Algebraic properties of the dot product,4. Orthogonal Vectors(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

14、 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.,5. Projections,6. The cross product,Definition 2.3.3: Let u and v are vectors, then the cross product of u and v, denoted uv,

15、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.,Unit vector, direction,7. Remember the form of the cross product (two methods),determinant,8. Algebraic properties of the cross produc

16、t,9. Geometric properties of the cross product,10. Triple products（三重积）,11. Tests for collinearity and coplanarity,2.3 Exercise P148 48,Theorem: Let u, v and w be nonzero three dimensional vectors. u and v are collinear if and only if uv=0. u, v and w are coplanar if and only if u(vw).,2.4 Lines and

17、 planes in space,1. The equation of a line in xy-plane,y,2. The equation of a line in three space,(1)Point and directional vector form equation of a line,(2)Parametric equations of a line,Example1: Let L be the through P0=(2,1,6), having direction vector u given by u=4,-1,3T. Find parametric equatio

18、ns for the line L. 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).,2. The equation of a plane in three space,Point and normal vector form equation of a plane,Example3: 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).,The relationship between two lines or two planes,