( 英 文 ) V ectors may also be m ult ip lied by a number.T he produ ct of th e vec tor a by t he n umb er is defined as t he vector a 一 a , th e absolute value of w hic h is obtained by m ultiply ing t he absolute value of t he vec tor口 by t he absol ute valu e of the number ,i.e.( a (== ( ( {a (,the direction coinciding with t he direct ion of t he vec tor a or being in t he oppos ite sense dependin g on whet her >0 or <二0 . If 一 0 or a = 0 .t hen a is consid ered t o be eq ual t o t he zero vect or. T h e m ult iplicat ion o f a vec to r b y a n u mb er p os sess es th e ass ociativ e Iaw and t w o d ist rib ut ive Iaw s.N amely .fo r any nu mb er s , and vecto rs a ,b ( 口) : ( ) 口( ass oc iat ive 1aw ) . +. 、。 。. 。I(distrib ti、,e· (a+ )===入口+ b f u Ve laws). L et us pro ve t h ese p rop er ties. T h e abso lu te val ues of t he vec to rs ( a ) an d ( ) a are the same and are equal to f f f f f口f.The directions of th ese vect or s eit her co incid e w it h t he d irect ion o f th e vect or a ,if a nd are of t he same sig n ,or being in t h e op pos it e s ens e, if an d h ave dif fer en t sig n s. H en ce ,t h e vect ors ( 口) a nd ( ) a are eq ual by ab so lu te val ue an d are i n t h e sam e direct ion ,consequ ent ly ,t h ey are equa 1. I f at 1east o ne of t h e num b ers , ’,or t h e vector a is equ al to zero ,th en bo t h vec to rs are equa 1 to zero an d ,h en ce ,t h ev are eq ual to each ot h er.T he as so ciativ e Iaw is t h us p rov ed. W e are no w go ing t o...