VECTORS VECTORS VECTORS VECTORS. 2. Vector Representation. 1. Definition. 3. Types of Vectors. 5. Vector Operations I. 4. Equal and Opposite Vectors

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1 1. Defnton A vetor s n entt tht m represent phsl quntt tht hs mgntude nd dreton s opposed to slr tht ls dreton.. Vetor Representton A vetor n e represented grphll n rrow. The length of the rrow s the mgntude of the vetor nd the rrowhed shows the dreton of the vetor. B Emple 1.1 A fore tng on od s vetor. Suh vetor represents oth the mgntude nd the dreton of ton. A The ove vetor s referred to s: AB or The mgntude s referred to s: AB, or, or smpl AB, or 3. Tpes of Vetors poston vetor AB (pont A s fed) lne vetor (t n slde long ts lne of ton) free vetor (not restrted n movement) Dr. E. Mlonds EX11/ 1 Dr. E. Mlonds EX11/ 4. Equl nd pposte Vetors Equl Vetors 5. Vetor pertons I 5.1 Addton the hn method drw the vetors strtng the seond where the frst ends (for two vetors). The sum s the vetor from the strt of the frst to the end of the seond. = { = nd nd hve the sme dreton} = + pposte Vetors In smlr w, for more thn two vetors = - { = nd nd hve opposte dreton} d d = + + Dr. E. Mlonds EX11/ 3 Dr. E. Mlonds EX11/ 4

2 5.1.1 Bs Propertes of Vetor Addton the prllelogrm method drw prllelogrm wth the two vetors s two dent edges hvng ommon strtng pont. The dgonl of the prllelogrm s the sum of the two vetors. + = + ( u+ v) + w = u+ ( v+ w) + = + = + ( ) = (5.1) 5. Sutrton θ - = + = + osθ = - Dr. E. Mlonds EX11/ 5 Dr. E. Mlonds EX11/ Multplton Slr The produt of vetor slr s vetor wth mgntude nd dreton the dreton of f > or the opposte dreton of f <. 6. Components of Vetor An vetors whose sum gves ertn vetor re the omponents of vetor -,, d re the omponents of vetor ( = + + d) d 7. Unt Vetors Propertes of Multplton Slr ( + ) = + ( + ) = + ( ) = ( ) 1 = = ( 1) = (5.) A vetor wth unt mgntude s unt vetor = = 1 Then = = - nd - re the oordntes of vetors nd wth respet to Dr. E. Mlonds EX11/ 7 Dr. E. Mlonds EX11/ 8

3 8. Crtesn Coordnte Sstem In three-dmensonl spe vetor v n e represented three omponents the most. ne w to do tht s to use set of three es of referene orthogonl to eh other so the omponents of vetor v le ross these es. Ths sstem of es s lled Crtesn oordnte sstem 8.1 Mgntude of Vetor = + + or = where,, re the unt vetors long the es of referene 1,, 3 re the oordntes of vetor wth respet to the gven sstem of referene = (8.1) Dr. E. Mlonds EX11/ 9 Dr. E. Mlonds EX11/ Dreton Cosnes of Vetor Emple 8.1 Let two vetors = + 3 nd = The mgntudes of nd re 1 α γ β 3 = + ( 1) + 3 = 14 = 374. = ( ) ( 5) = 38= 616. The dreton osnes of re l = = m= , = 7., n= = The dreton osnes l m n re the osnes of the ngles etween the vetor nd the es of referene. l m 1 3 =, =, n = l + m + n = 1 (8.) The sum of nd s + = = ( ) + ( 1+ 3) + ( 3 5) = + Dr. E. Mlonds EX11/ 11 Dr. E. Mlonds EX11/ 1

4 9. Vetor pertons II 9.1 Slr or Dot Produt The dot produt of two vetors nd s slr quntt epressed s nd gven =osθ (9.1) Propertes of Slr Produts q+ q = q + q 1 1 = = ff = ( + ) = + (9.) 9.1. Slr Produts of the Unt Vetors n n rthogonl Sstem θ If θ = 9,.e. the two vetors re perpendulr, then osθ = nd =. Vetors nd re lled orthogonl. Emple 9.1 Wor of fore. It s es to prove from (9.1) nd (9.) tht = = = 1 = = = = = = (9.3) Dr. E. Mlonds EX11/ 13 Dr. E. Mlonds EX11/ Slr Produts n Crtesn Coordntes If = + + nd = + +, then = ( + + ) ( + + ) Proeton of Vetor on to Another Vetor Usng propertes (9.) nd (9.3) we hve = (9.4) Mgntude nd Angle n Terms of Slr Produt Aordng to (9.1) = os( ) =. Therefore θ p = (9.5) Also from (9.1) nd (9.5) the ngle etween two vetors nd s p= osθ U p = V W = osθ (9.8) osθ = = (9.6) If l1, m1, n1 nd l, m, n re the dreton osnes of nd respetvel t n e esl shown from (8.) nd (9.6) tht osθ = ll 1+ mm 1 + nn (9.7) 1 Dr. E. Mlonds EX11/ 15 Dr. E. Mlonds EX11/ 16

5 Emple 9. Consder the two vetors from emple 8.1 = + 3 nd = Then = ( ) + ( 1) 3+ 3( 5) = 9. Vetor or Cross Produt The ross produt of two vetors nd s vetor v epressed s suh tht v = = snθ (9.9) nd the dreton of v s perpendulr to oth nd usng the rghthnd rule. From emple 8.1 = nd = The ngle etween the two vetors s v osθ= = = 96. θ= θ The proeton of to s p = = = Also, f = + + nd = + +, then = (9.1) Dr. E. Mlonds EX11/ 17 Dr. E. Mlonds EX11/ 18 Emple Propertes of Vetor Produts Moment of fore Emple 9.4 Consder gn the two vetors from emple 8.1 = + 3 nd = Then ( l) = l( ) = ( l) ( + ) = ( ) + ( ) ( + ) = ( ) + ( ) = ( ) 9.3 Slr Trple Produt The slr trple produt of three vetors,, s the slr (9.11) Note tht v = = = = = It n e shown tht v = ( ) (9.1) ( ) = (9.13) v = v = It n e shown tht the slr trple produt s the volume of the prllelepped wth,, s edge vetors. Dr. E. Mlonds EX11/ 19 Dr. E. Mlonds EX11/

6 1. Lnes nd Plnes 1.1 Lnes n 3-d Spe (Prmetr Epresson) Emple 1.1 () l r r n r t The strght lne tht psses through nd s prllel to the unt vetor r = 3 1 n = 1 s gven r() t = ( 3+ t) + Gven re one pont r of the lne nd the dreton of the lne through the unt vetor n. Then, for ever pont r r = r + r = r + tn t = ( r + tn ) + ( r + tn ) + ( r + tn ) (1.1) where t s free prmeter. Dr. E. Mlonds EX11/ 1 Dr. E. Mlonds EX11/ 1. Plnes 1..1 Plne Defned ts Dstne from the rgn 1.. Plne Defned ts Dreton nd Pont n r r r If = s the dstne of the plne from the orgn, then ever vetor r = + + tht ponts to the plne hs the sme proeton = n the dreton of. Aordng to (9.8) = r r = = 1 Hene + + = = (1.) Suppose tht the plne psses through the pont r = r + r + r nd t s perpendulr to the unt vetor n= n+ n + n. Then for ever pont r = + + on the plne.e. Hene r r s perpendulr to n ( r r ) n= r n= r n= n + n + n = r n= (1.3) Dr. E. Mlonds EX11/ 3 Dr. E. Mlonds EX11/ 4

7 Emple 1. Fnd the equton of the plne tht psses through the pont r = r + r + r = nd s perpendulr to the unt vetor 11. Curves, Tngents, Velot nd Aelerton 11.1 Curves (prmetr epresson) C 1 1 n= n+ n+ n = For ever pont r = + + s vld n + n + n = r n e F + = H 6 or = I K r( t) Hene the plne equton s + = 8 A urve C n e represented prmetrll vetor funton r() t = () t + () t + () t The end of r(t) moves long C s the slr prmeter t hnges. Dr. E. Mlonds EX11/ 5 Dr. E. Mlonds EX11/ 6 Emple Tngent to Curve The vetor funton u( t) Q r() t = Rosωt+ Rsnω t P r r( t + t) hs mgntude r( t) = R os ωt + R sn ω t = R r( t) Hene r(t) represents rle of rdus R nd entre t the orgn of the plne. C r = r( t + t) r( t ) r( t) Then the vetor dr r r + r r ( ) = = lm = lm ( t t t ) ( t ) (11.1) dt t t t t s the dervtve of r(t) wth respet to t nd s the tngent to the urve C t the pont P. u(t) s the unt vetor n the dreton of r () t nd s lled the unt tngent vetor to the urve C t the pont P. Hene 1 u() t = r () t r ( t) (11.) Dr. E. Mlonds EX11/ 7 Dr. E. Mlonds EX11/ 8

8 Emple Velot Consder the poston vetor r() t = () t + () t + () t tht defnes urve C. If t s the tme then the dervtve r () t s the velot of prtle tht moves long C. Aordng to (11.1) r () t s tngent to the urve C nd ponts to the dreton of moton. The vetor funton r() t = Rosωt+ Rsnω t defnes the movement of prtle long rle of rdus R nd entre t the orgn of the -plne wth onstnt ngulr velot ω. The velot s gven nd v() t = r () t = Rω snωt+ Rω osω t dr d d d v() t = r () t = = + + (11.3) dt dt dt dt The elerton then s v = v = v v = Rω () t = v () t = Rω osωt Rω snωt= ω r 11.3 Aelerton Usng smlr rguments the elerton vetor (t) s gven v( t) d r d d d () t = v () t = = + + (11.4) dt dt dt dt ( t) Dr. E. Mlonds EX11/ 9 Dr. E. Mlonds EX11/ 3