Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m


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1 . Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion L= Projetion vetor AC = ( AB AD AB AD ) AD Foot of perpeniulr = OC =OAAC Shortest istne from B to line A C D = BC = AB L L [OR = AB AD ] (ii) Aute ngle etween two lines: m m θ = os where m n m re the iretion vetors of the m m the two lines. (iii) Aute ngle etween two plnes: n n θ = os where n n n re the iniviul normls to the two n n plnes respetively. (iv) Aute ngle etween line n plne: π m n θ = os where m n n re the iretion vetor of the line m n n norml to the plne respetively, n θ is the ngle etween the line n plne in question.
2 . Cross prout (vetor prout): =[ sinθ] n where n is vetor tht is perpeniulr to oth n. Lws of ross prout: (i) =( ) (iii) = ~ (ii) ( ) = =( ) ( ) Applitions: (i) Are of tringle= (ii) If four points A, B, C n D re oplnr, then AB AC AD=. Eqution of lines: Representtions: (i) r = λ e (prmetri form) OR r = λm (onense form) f x y (ii) = = (rtesin form) e f. Equtions of plnes: Representtions: (i) r = λ m µ m (prmetri form) (ii) r n= n (slr prout form) (where n = m m, is the position vetor of point lying on the plne. ) (iii) x y = k (Crtesin form) (where, n re the omponents of the norml vetor to the plne) 5. Skew lines: Two lines with equtions r = λm n r = µ m re si to e skew lines if they DO OT interset t ommon point n m is OT PARALLEL to m. 6. Determining if line resies in plne: A line with eqution r = λm is si to resie in the plne r n= k if (i) m n= (ii) n= k
3 . Shortest istne from plne to origin: For plne with eqution r n= k, the shortest istne from the plne to the k origin is given y. n 8. Distne etween plnes: For plnes with equtions r n= k n r n= k, where k < k, the shortest istne etween them is given y: (i) k n k n if k n k re of ifferent signs (ii) k n k n if k n k re of the sme signs 9. Fining intersetion etween vrious onstruts: (i) Intersetion etween lines: For lines with equtions r = λm n r = µ m, equte them to eh other in olumn vetor form suh tht λm = µm. Solve for the vlues of λ n µ efore sustituting k into either of the two line equtions to erive the ommon point of intersetion. (ii) Intersetion etween line n plne: For line with eqution r = λm n plne with eqution r n= k, sustitute the line eqution within tht of the plne eqution suh tht ( λ m) n= k. Solve for the vlue of λ n susequently erive the ommon point of intersetion through sustitution of λ into the line eqution. (iii) intersetion etween plnes: A. If one plne is presente in slr prout form n the other in prmetri form, Exmple: r = () r = λ µ () λ µ λ µ = 6 λ
4 λ µ 9λ µ λ = 6 λ µ = λ µ = µ = λ Sustituting this k into (), Eqution of line of intersetion is r = λ ( λ) = λ (shown) B. If oth plnes re presente in Crtesin form: Exmple: x y = () xy = () ()(): = = 5 Let y= t n sustituting this together with = 5 into (), We hve x= t Eqution of line of intersetion is x t r = y= t = t (shown) 5 5 C. If oth plnes re presente in slr prout forms or in prmetri forms or one is presente in slr prout form n the other in prmetri form, onvert the plne equtions suh tht their onfigurtions mthes tht of either se A or B, n solve oringly. D. If ommon point A with position vetor is known to resie on oth plnes, n the two plnes hve norml vetors n n n, then the ommon line of intersetion is simply given y r = λ ( n n ). (iv) Intersetion etween plnes: Extrt the omponents of the seprte plne equtions to form the ugmente mtrix:
5 r =, r = r = After reuing the ugmente mtrix to its row reue equivlent using the RREF funtion of the grphi lultor, possile senrios rise: A. The plnes interset t one point, ie there is unique solution to the mtrix. Exmple: (x) (y ) ( )=, therefore x =, (x) (y ) ( )=, therefore y =, (x) (y ) ( )=, therefore = Hene, the plnes interset t the point (,,). B. The three plnes o not interset t ll. Exmple: For the thir row in the reue form mtrix, =, giving rise to ontrition, hene there is no ommon point to the plnes, ie they DO OT interset.
6 C. The three plnes interset t line. Exmple: From the reue row mtrix, we hve x x = =, y y = = Let, =λ then r = y x = = λ λ Therefore, the three plnes interset t the line r= t, where R = λ t
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