1、Section 8.212The Dot Product(点积点积,数量积数量积,内积内积)of Two VectorsThe dot product can be used toexpress the work done by a givenforce.Let a and b be two vectors,and suppose is the angle Definition Then the real numbera,b.between a and b,denote bya b,is called the dot product(scalar product,inner product)o
2、f a and b,denoted byaba b|a|(b)|b|(a).that is,|a|b|cos ora b|a|b|cos 3The Basic Properties of Dot Productand a a0a0;2a a|a|0,(2)Nonnegativity:a(bc)a ba c,(3)Commutative law:a bb a;(5)Associative law with the scalar multiple:(a)b(a b);kk(4)Distributive law:(ab)ca cb c,The dot products have the follow
3、ing basic properties:00 aa 0 (1)4The Component Representation of The Dot ProductSince the basic unit vectors i,j,k are perpendicular to one another,using the definition of the dot product we haveIfi jj kk i0.i ij jk k1 andandbijkxyzbbbaijkxyzaaathena b(ijk)(ijk)xyzxyzaaabbb.xxyyzza ba ba bExamples64
4、28.(1)(6)(2)(2)(2)(1)(1,2,2)(6,2,1)(1)231.(2)1i3j4ij21(4)(3)(1)25Some Applications of The Inner Product in Geometry(1)The norm(模，范数模，范数)of a vectorBy the definition of the norm of a vector,we have|a|a a 222.xyzaaa(2)The included angle(夹角夹角)between two nonzero vectorsLet a and b be two nonzero vector
5、s.We havea b|a|b|cos a bcos,|a|b|222222cosxxyyzzxyzxyza ba ba baaabbb a b ab0 xxyyz za ba ba b6Some Applications of The Inner Product in Geometry(3)The Projection(投影投影)then we haveaba b|a|(b)|b|(a),Since ba b(a)|b|a b andaa b(b)|a|b a and the projection vector of a onto b isba bproj ab(a b)b.|b|ther
6、efore the projection vector of b onto a isaa bproj ba(b a)a.|a|7Some Applications of The Inner Product in GeometryExample andb(1,2,2).Suppose that a(1,1,4)(1)Find a b(2)Find the angles between a and b(3)Find the projection of a onto b.Solution(1)a b 1 11(2)(4)2 9.222222(2)cosxxyyzzxyzxyza ba ba baaa
7、bbb 1,2 3.4 b(3)a b|b|(a)ba b(a)3.|b|Finish.Some Applications of The Inner Product in Geometry8Example and(1,2,2),D For the points(1,4,3),(3,1,2),(6,1,9)ABCprove that the line through A and B is perpendicular to the line through C and D.Solution(31,14,23)(2,3,1).AB (16,21,29)(5,1,7).CD (2,3,1)(5,1,7
8、)10370.AB CD So,the line through A and B is perpendicular to the line through C and D.Finish.9Some Applications of The Inner Product in GeometryExample Prove the Cauchy inequality:,(1,2,3);iia bR iLet112233322111i iiiiiia bab Proofand123b(,).b b b Let the vector123a(,)a a a Since a b|a|b|cos,we have
9、|a b|a|b|.By the component representation of dot product,we will obtain theCauchy inequality.Finish.10The Vector Product of Two Vectors in SpaceWe start with two nonzero vectors u and v in space.If u and v are notparallel,they determine a plane.We select a unit vector n perpendicularto the plane by
10、the right-hand rule.This means that we choose n tobe the unit(normal)vector that points the way your right thumb points when your fingers curl through the angle from u to v.can be defined uv Then the vector product as following.11Definition Vector(Cross,outer)Product(向量积向量积,叉积叉积,外积外积)sinuv(|u|v|)nTh
11、e vector product of u and v is often called the crossmultiple of n.uv.product of u and v because of the cross in the notationis orthogonal to both u and v because it is scalarThe vectoruv Since the sines of 0 and are both zero,it makes sense to define thecross product of two parallel nonzero vector
12、to be 0.to be zero.If one or both of u and v are zero,we also defineuv The Vector Product of Two Vectors in Space12The Vector Product of Two Vectors in SpaceProof Two vectors a and b are parallel(or collinear)if and only if We assume that a and b are both nonzero vectors.,If a=0(or b=0),then this co
13、nclusion obviously holds.(,)0a b or0.a b Theoremsin(,)0.a b /0.aba b that is,Two vectors aand b are parallel(or collinear)if and only if We have that,13Properties of the Vector ProductIf u,v and w are any vectors and r,s are scalars,then(1)Anti-commutative law u vv u;(3)Distributive law(vw)uv uw u.u
14、(vw)u vu w;(2)Associative law with respect to the scalar multiple(u)(v)()(u v);ssrr(u)vu(v)(u v);rrr14The Component Representation of the Vector ProductThen the distributive laws and the rules for multiplying i,j,and123123uijk,vijk.uuuvvvSuppose thatk tell us that123123u v(ijk)(ijk)uuuvvv1 11 21 32
15、12 22 33 13 23 3i iiji kj ij jj kk ikjk ku vu vu vu vu vu vu vu vu v 2 33 21 33 11 22 1()i()j()k.u vu vu vu vu vu v233213311221uv()i()j()k.u vu vu vu vu vu v15The Component Representation of the Vector Product233213311221uv()i()j()ku vu vu vu vu vu vThe terms in the last line are the same as the ter
16、ms in the expansionof the symbolic determinant123123ijk.uuuvvv16The Component Representation of the Vector ProductSolutionIfv4i3j k,andu2ij k andv u.findu v ijkuv211431 112121ijk3141432i6j10k.vu(uv)2i6j10k.Finish.17The Component Representation of the Vector ProductSolutionLet be the angle between th
17、e vectorsv2ij2k,andu6i2j3kfindijkuv623212Example 236362ijk1222217i6j10k.222(7)(6)(10)|uv|37sin.|u|v|77 5 Finish.sin.18The Geometric Meaning of the Norm of the Vector Productisu v Because n is a unit vector,the magnitude of|u v|u|v|sin|n|u|v|sin.This is the area of the parallelogramdetermined by u an
18、d v,|u|being thebase of the parallelogram and|v|sin|the height.sinu v(|u|v|)n19The Component Representation of the Vector ProductSolutionFind the area of the triangle whose vertices are(1,1,1),(2,0,1),(1,1,3).ABC Since bparallelogram with adjacent sides AB and AC,that isThe area S of the triangle AB
19、C is half the area of the224,ijk 144166.2S Example 1|2SABAC(1,1,0)(2,2,2)ABAC 11022 2ijk Finish.We have 20 Vector ProductExample Find the line speed vector v at any point P on the body.SolutionWe take a vector w in the axis l,suchand its positive direction is determined|,thatLet O be a point on the
20、axisby the right-hand rule.l,then the line speed0|v|.P P uuu vLetr;OP uuu vThe line speed vector v0|r|sin(,r).P P uuu vthenis perpendicular to both w and r,and w,r,v satisfiesTherefore the right-hand rule.vr.A rigid body rotates around a fixed axis l with angular velocity w.Finish.21TorqueWhen we tu
21、rn a bolt by applying a force F to a wrench,the torque weproduce acts along the axis of the bolt to driveThe magnitude of the torquethe bolt forward.depends on how far out on the wrench theforce is applied and on how much of theforce is perpendicular to the wrench at theThe number we use topoint of
22、application.measure the magnitude is the product of thelength of the lever arm r and the scalar componentof F perpendicular to r.22TorqueIn the right figure,we haveIt is well known that if the force F is parallelto the wrench,the torque produced is zero.|r|F|sin,Magnitude of torque vector|rF|.orIf w
23、e let n be a unit vector alongthe axis of the bolt in the direction of thetorque,then a complete description of theorTorque vectorrF,torque vector is(|r|F|sin)n.23Triple Scalar or Box ProductDefinition Triple Scalar or Box Product(uv)w Suppose u,v and w are three vectors,then the productis called th
24、e triple scalar product of u,v and w.Sometimes the triple scalar product can be denoted byabc(ab)c.|(uv)w|uv|w|cos|,It is easy to see from the formulathe absolute value of the product is the volume of the parallelepipeddetermined by u,v and w.24The Geometric Meaning of the Triple Scalar Product25The
25、 Component Representation of The Triple Scalar ProductThe triple scalar product can be evaluated as a determinant:231312123231312uuuuuuwwwvvvvvv231312231312(uv)wijkwuuuuuuvvvvvv123123123.uuuvvvwww 26The Properties of Triple Scalar ProductBy means of the properties of the determinants,it is easy to obtain thefollowing properties of the triple scalar product:(uv)w(vu)w.(2)(1)(uv)w(vw)u(wu)v.(3)u,v and w are coplanar if and only if(uv)w0.Triple Scalar Product27Find the volume of the parallelepiped whose edges are,23,2.aijk bij cij ()Vabc123123123111()230210aaaab cbbbccc Solution238.21 8.V
