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1、11.4 The Cross ProductProblem IntroductionHow to define the product of the two vectors in another way?DefinitionNow we introduce the cross product(or vector product);it will also have many applications.123123The cross product of,and,is defined byuvuu u uvv v v=The anticommutative law:()u vv u=2 33 2
2、1 33 11 22 1,uvu vu vu vu v u vu v=+231312123231312123Using determinants,we can write the definition of asuvijkuuuuuuuvuuuijkvvvvvvvvv=+Theorem AGeometric Interpretation of uv:Let u and v be vectors in three-dimensional space and ab a bright-handed system(1)()0(),that is,is perpendicular to both and
3、(2),and form a right-handed system;(3)|=|sinu uvv uvuvuvu vuvuvu v=；Theorem BThat is a simple consequence of Theorem A.Two vectors u and v in three-dimensional space are parallel if and only if uv=0.DefinitionAttention：0)1(=aa)0sin0(=Vector product is also called cross product or outer product.1.Def
4、inition:and abCross product of vectorsis cab=|sincab=Length:included angle of,a bis perpendicular to both andand follows right-handed system.ca,bDirection:ba)2(/0=ba)0,0(baDefinition2.Cross product follows the following laws:PROOF,0sin=,0 =ba/)(0 or=0sin=ba/;ab =ba(1)Anti-commutativity law)(,0|a,0|b
5、0=ba(2)Distributive law=+cba)(;cbca+ba)2(/0=ba)0,0(ba=|ba=sin|ba.0=ba)(=)(ba).(ba (3)If is a scalar DefinitionDoes cross product of vectors satisfy cancellation law?No!Generally,No.attention,caba=.cb=0 aattentionDoes cross product of vectors satisfy commutative law?Definition3.Calculate cross produc
6、t using coordinates expression,kajaiaazyx+=kbjbibbzyx+=Let=ba)(kajaiazyx+)(kbjbibzyx+kbabajbabaibabaxyyxzxxzyzzy)()()(+=coordinates expression of cross product Distribution law 0iijjkk=ijk=jki=kij=jik=kji=ikj=Definitioncross product can also be expressed by determinants zzyyxxbababa=We getba/0=bazyxzyxbbbaaakjiba=0=zyxbbb,can not be 0 simultaneously.|ab=|sinabIn fact,Example 1Find the unit vector that is perpendicular to vectors324,aijk=+2bijk=+=kj5152xyzxyzijkcabaaabbb=324112ijk=105jk=+22|1055 5c=+=0|ccc=SummaryThe definition of cross productOperation lawsTheorems A and BThe Cross Product