1、Section 8.11Vectors2ScalarsSome of the quantities these we measure are determinedwe need only write down a number and name and appropriate unit of measure.The associated real numbers are scalars(数量数量或或标量标量).by their magnitudes.Vectorsforce,displacement,etc.ExampleExample length,area,mass,temperature
2、,etc.They are represented bySome of the quantities these we measure are determined bytheir directions and magnitudes.directed line segments(有向线段有向线段).These quantities arecalled vectors(向量向量或或矢量矢量).Vectors Definition Vectors are quantities that possess both magnitude and direction.3magnitude(or lengt
3、h)&directionInitial pointOPaTerminal point Definition The magnitude of a vector is called the length or norm(模模)of the vector.The length of vectors and a is written as and|a|,respectively.OP OP|A vector is usually represented by a directed line segment,denoted by or .aOP Vectors4Definition Two vecto
4、rs are equal(相等的相等的)or the same if they have the same magnitude and direction.Definition The negative vector(负矢量负矢量)is a vector which has the same magnitude with the original b,and has an opposite direction with b,denoted by b.-bbaVectors5 Definition If two vectors a and b have same or oppositedirec
5、tion,we say that they are parallel(平行平行)or collinear(共线共线),denoted by a/b.a common initial point.If they line in the same plane,then we say Definition Suppose that a1,a2,ak(k2)are k vectors with that these vectors are coplanar(共面共面).Definition If the directions of two vectors a and b are orthogonal,
6、we say that they are orthogonal(正交正交)or perpendicular (垂直垂直),denoted by ab.NoteVectors Definition A vector whose length is 1 is called a unit vector(单单位向量位向量),a unit vector whose direction is the same as that of a is written as a。.Definition A vector whose length is 0 is called the zero vector(零向量零向
7、量),and is written as 0.6In textbooks,vectors are usually written in lowercase,boldface letters,In handwritten form,it is customary Sometimes we use uppercase boldface letters,w.for example,u,vandF,such asto denote a force vector.and.F to draw small arrows above the letters,for example,u v w Operatio
8、ns on Vectors Definition Triangle law of addition of vectors(向量加法的三向量加法的三角形法则角形法则)Suppose a and b are two vectors.If we draw a vector,which is equal to b,from the initial point of a,then the sum a+b of a and b is the vector extending from the terminal point of a to the initial point of b.7a+bOAaB bO
9、Bab Operations on Vectors Parallelogram law of addition of vectors(向量加法的平行四向量加法的平行四边形法则边形法则)8Cba+bBOAaOBab Operations on Vectors9The addition of vectors satisfies the following laws:(1)Commutative law abba;(2)Associative law(ab)ca(bc);(4)a(a)0.(3)a0a;abb+aacb+cbba+ba+b+ca+baOperations on Vectors10De
10、finition The difference(减法减法)of two vectors a and b is given byaba(b).ba-b-ba-ba Definition Scalar Multiplication(数乘向量数乘向量)The product of a scalar m and a vector a expressed by ma is a vector.Its length is|ma|.Its direction is the same as that of a if m is positive and is opposite to that of a if m
11、is negative.la(l0)ma(m 0,so that the denominator of theThusabove fraction is negative.11cos.212 1 and121cos,cos,cos,222 Finish.Example Suppose that a set of direction numbers of the vector a1,2,1,is and the included angle between a and the Oz axis isan acute angle.Find the unit vectora.33Angle betwe
12、en two vectorsTaking any pointincluded angle(夹角夹角)between the vectors aIf the included angle between a and b isThen the angleSuppose that a and b are any two nonzero vectors.,.MAaMBbuuu vuuu v M in the space,we can draw vectorsaABM(not greater than)is called theAMB a,b.and b,denoted bythen a and b a
13、re said to,be,2 denoted byab.b34Suppose that a and b are any two nonzero vectors which are notThe Projection of Vectors（投影向量）（投影向量）terminal point B2 of the vector brespectively,such that both of themare perpendicular to the orientedstraight line which the vector a lieson and has the same direction a
14、spoints of the planes and the straightDraw two planes through the initial point B1 and theperpendicular.a.be the intersectionLet1B and2B line;is called12B B uuuu vthen the vectorthe projection vector of b onto a,denoted byaproj b.It is easy to see thataproj b|b|cos a.35The Projection of VectorsDefin
15、ition Orthogonal projection vector,orthogonal projectionThen the vectoris called the orthogonal projection vector of b onto a(or onto the unit vectorwith the same direction as a),or simply the projection vector(投影向量投影向量).The scalarwith the same direction as a),or simply the projection(投影投影).is calle
16、d the orthogonal projection of b onto a(or onto the unit vector Suppose that the included angle between the vectors a and b is.a(b)|b|cos aproj b|b|cos a 36aThe Projection of VectorsLet the vectorThis formula shows that the vectorb can be decomposed into the sumabproj bv.andavproj b thenbaproj bavbp
17、roj bavbproj b,v.and of two perpendicular vectorsaproj bcomponent of b Orthogonal to a.is called the component of b along a and v is called theThis is called the orthogonal decomposition(正交分解)of b;aproj b37The Projection of VectorsThe first formula is obviously true from the definition of the projection.We haveTherefore,aaa(bc)(b)(c).The second formula can be proved as following.aa(b)(c)|b|cos|c|cos,OBB COC a(bc)|bc|cos,OC Properties Projections have the following properties(where a,b,care vectors,k is a real number):(1)aa(b)(b);kk(2)aaa(bc)(b)(c).(b)|b|cosa
