Transcription of LECTURE 2: VECTOR MULTIPLICATION - SCALAR AND …
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LECTURE 2: VECTOR MULTIPLICATION - SCALAR AND …
LECTURE 2: VECTOR MULTIPLICATION - SCALAR AND VECTORPRODUCTSProf. N. HarnewUniversity of OxfordMT 20121 Outline: 2. VECTOR SCALAR Properties of SCALAR Angle between two VECTOR Properties of VECTOR VECTOR product of unit VECTOR product in Geometrical interpretation of VECTOR SCALAR ProductScalar (or dot) product |a|.|b|cos abcos (write shorthand|a|=a ).IScalar product is the magnitude ofamultiplied by the projection ifais perpendicular |a|2(since =0 )Hencea= ( ) Properties of SCALAR product (i) (ii) This leads (axi+ayj+azk).(bxi+byj+bzk)=axbx+ayby+az bz(this is a very useful relation)iii) : commutativea.(b+c) = + : distributive(iv) Ifc=a+bThenc2= (a+b).(a+b)=a2+b2+ +b2+2abcos( ab)(v) Parentheses are importantNote( )w6=u( )because one is a VECTOR along w,the other is along Angle between two vectorsBy definition cos( ) = angle between vectorsa= (3,1,5) andb= (2,1,3)Icos =3 2+1 1+5 3 (32+12+52) (22+12+32)=22 (35) (14)= = Example of SCALAR products in physicsIWork done on a body by a force through distancedxIdW = the component of force parallel to displacement VECTOR ProductVector (or cross) product of two vectors, definition:a b=|a||b|sin nwhere nis aunit vectorin a directionperpendicularto get direction ofa buse right hand rule:Ii) Make a
2.1 Scalar Product Scalar (or dot) product definition: a:b = jaj:jbjcos abcos (write shorthand jaj= a ). I Scalar product is the magnitude of a multiplied by the projection of b onto a.
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