LECTURE 2: VECTOR MULTIPLICATION - SCALAR AND …

1 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.

2 Ii) Make a set of directions with yourrighthand thumb & first index finger, and withmiddle finger positioned perpendicular toplane of bothIii) Point your thumb along the first vectoraIiii) Point your 1st index finger alongb,making the smallest possible angle toaIiv) The direction of the middle finger givesthe direction ofa Properties of VECTOR productI(a+b) c= (a c) + (b c): distributiveIa b= b a: NON-commutativeI(a b) c6=a (b c): NON-associativeIIfmis a SCALAR ,m(a b) = (ma) b=a (mb) = (a b) b=0 if vectors are parallel (0o) a= VECTOR product of unit vectorsThe basis vectors are connected bycyclic permutations of VECTOR products(another good way to remember theright hand rule)Ii j=kIj k=iIk i= VECTOR product in componentsA very useful property:Ia b= (ax,ay,az) (bx,by,bz)= (axi+ayj+azk) (bxi+byj+bzk)ISincei i=j j=k k=0 andi j= b= (aybz azby)i (axbz azbx)j+ (axby aybx)kThis is much easier when we write indeterminantform:a b= ijkaxayazbxbybz.

3 (1) Geometrical interpretation of VECTOR productVector product is related to the areaof a triangle:IHeight of triangleh=asin IArea of triangle =Atriangle=1/2 base height=bh2=absin 2=|a b|2 IVector product therefore givesthe area of the parallelogram:Aparallelogram=|a b|IHence VECTOR area Aparallelogram=a bwhere thevector points perpendicular to theplane of the ExamplesExample 1 Find the area of a parallelogram defined by coordinates (0,0,0),(1,3,4) and (2,1,3).IMake vectorsa= (i+3j+4k)andb= (2i+j+3k)a b= ijk1 3 42 1 3 .(2)Ia b= (3 3 4 1)i (1 3 4 2)j+ (1 1 3 2)k= 5i+5j 5kIThus the area is (52+52+52) = method certainly beats 1/2 base height !11 Example 2 Example of scalars and cross product :Show that ifa=b+ cfor some SCALAR , thena c=b :a=b+ c a c= (b+ c) c=b c+ c cIbutc c=0 Isoa c=b cQED12 Examples of VECTOR products in PhysicsIa) TorqueAtorqueabout O due to a forceFacting at B :T=r F.

4 Torque is avector with direction perpendicular tobothrandF, magnitude of|r||F|sin .Ib) Angular momentumA body with momentumpat positionrhas angular momentumabout O ofL=r p. Angular momentum is a VECTOR withdirection perpendicular to bothrandp, magnitude of|r||p|sin .Ic) Lorentz forceThe force exerted by a magnetic fieldBon a chargeqmovingwith velocityvis given byF=qv B13