E.g. Find the ratio of intersection of the medians of a triangle. ksoiqk ffoYsl Ndú;fhka ;%sflaKhl uOHia: fPaokh jk wkqmd;h fidhkak. A B C s 1 t 1 D E.

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Presentation transcript:

E.g. Find the ratio of intersection of the medians of a triangle. ksoiqk ffoYsl Ndú;fhka ;%sflaKhl uOHia: fPaokh jk wkqmd;h fidhkak. A B C s 1 t 1 D E ABC ;s%fldaKhg m%fuÞhh fhoSfuka F ABE ;%sfldaKhg m%fuÞhh fhoSfuka fuS ksid wkqrEm ffoYslhkays ix.=Kl ie,lSfuka 1 2

1 2 1 uOHia: 1:2 wkqmd;hg fPaokh fjS" A B C s 1 D E F kuS

;%s;aj.=Ks;h'/ Triple Product. ffoYsl ;=kla w¾:j;a f,i ;s;a.=Ks;h iy fyda l;sr.=Ks;h iu. fhoSfuka ;%s;aj.=Ks; w¾: olajkq,nhs. The triple products are defined using dot product and cross product meaningfully with three vectors. ;%s;aj.=Ks; / Triple Product ;%s;aj wosY.=Ks;h / Triple scalar product. ;%s;aj ffoYsl.=Ks;h/ Triple vector product.

;%s;aj wosY.=Ks;h / Triple scalar product. ffoYsl ;=ku tlu,laIHfhka wdruSN jk f,i w|suq' OA C D fujeks ;%sudk rEmhlg iudka;rKslrKh ( parallelopipe) hkak Ndú; fjS" ABCD

OA C D iudka;rKslrKfha mrsudj = ABCD. c ;jo tkuS ABCD = iudka;rKslrKfha mrsudj volume of the parallelopiped fus ksid ;jo

;%s;aj ffoYsl.=Ks;h / Triple vector product. X Y Z

Eq 1 Eq 2 So

E.g. Evaluate Let

f¾Ldjl iïlrKh" ( Equation of a line. ) A P O AP = fuh l g iudka;r ksid Vector form of the equation of a line. f¾Ldjl iïlrKfha ffoYsl wdldrh'

f¾Ldjl iïlrKh Symmetric form of the equation. f¾Ldjl iïlrKfha iuñ;sh wdldrh"

Cartesian form of the equation. f¾Ldjl iïlrKfha ldÜishdkq wdldrh'

f¾Ldjl iïlrKh igyk l hkq tall ffoYslhla kus g osYd fldaihsk hehs lsh;s' l hkq tall ffoYslhla fkdjk úg g osYd wkqmd; hehs lsh;s" direction cosines direction ratios

ksoiqk ( 1, 2, 0 ) iy ( 2, 3, 1 ),laIH Tiafia we;s f¾Ldfjs iïlrKh fidhkak' osYd wkqmd; iy osYd fldaihsk fidhkak" A P O A ( 1, 2, 0 ) iy B ( 2, 3, 2 ) hehs.ksuq" f¾LdfjS iïlrKh

f¾LdfjS iïlrKfha iuñ;sh wdldrh' f¾LdfjS iïlrKfha ldÜishdkq wdldrh' ksid osYd wkqmd; ( 1, 1, 1 ) fjs" osYd fldaihsk

ksoiqk ( 1, 3, 3 ) iy ( 2, 1, 0 ),laIH Tiafia we;s f¾Ldfjs iïlrKh fidhkak" A B A ( 1, 3, 3 ) iy B ( 2, 3, 0 ) hehs.ksuq. f¾LdfjS iïlrKh

f¾LdfjS iïlrKfha iuñ;sh wdldrh' f¾LdfjS iïlrKfha ldÜishdkq wdldrh'

f¾Ld foll fPaokh / Intersection of two lines iy f¾Ld fPaokh fjS kuS jk f,i wosY mej;sh hq;=hss' fuúg If the lines and intersect, there exist scalars such that ffoYsl iïlrKh ;Dma; jk f,i wosY mej;sh hq;=hss There exists scalars satisfying the vector equation

ksoiqk iy u.ska fokq,nk f¾Ld fPaokh fjS oehs fidhkak" Find the lines given by and intersect f¾Ld fPaokh fjS kus / If the lines intersect iïlrKh ;Dma; jk f,i wosY mej;sh hq;=hss tl úg ;Dma; jk f,i mej;sh hq;=hss

fuúg wi;H fjs" iïlrK ;Dma; jk f,i wosY fkdmj;S f¾Ld fPaokh fkdfjS. Since there do not exist scalars satisfying above three equations, the given lines do not intersect.

( 2, 1, 3 ) yryd jQ ( 1, 2, 5 ) osYd wkqmd; iys; f¾Ldfjs iïlrKh fidhkak" E.g.Find the equation of the line passes through the point ( 2, 1, 3 ) whose direction ratios is ( 1, 2, 5 ). f¾LdfjS iïlrKh úiªu fus ksid f¾Ldfjs iïlrKh u.ska fokq,nk f¾Ldj iy by; f¾Ldj fPaokh fjS oehs fidhkak"

Find whether the ine given by and the above line intersect by; f¾Ld fPaokh fjS kus 1 2 ;Dma; jk f,i mej;sh hq;=hss 3

2 1 3 fuúg i;H fjs' iïlrK ;Dma; jk f,i wosY mj;S tksid f¾Ld fPaokh fjS" fPaok,laIHh i|yd fPaok,laIHh i|yd ( 0, -3, -7 ) fjs' 1 2 3

ú;, f¾Ld /Skew lines p;=ia;,hla i,lkak. Consider a tetrahedron. A B C D AB iy CD, AC iy BD AD iy BC AB and CD, AC and BD, AD and BC are skew. ú;, fjs. m%;sM,h /Result

ksoiqk iy u.ska fokq,nk f¾Ld ú;, oehs fidhkak' /Find whether above lines are skew. A B f¾Ld tal;, kus tal;, fjs'

fus ksid f¾Ld tal;, fkdjk fyhska f¾Ld ú;, fjs / Since they are not coplanear they are skew.

;s%udkfha f¾Ld Lines in 3d ú;, f¾Ld / skew lines tal;, f¾Ld / coplanar lines iudka;r f¾Ld / parallel lines fPaokh jk f¾Ld / intersecting lines

;,hl iïlrKh" ( Equation of a plane. ) O AP = fuh n g,usnl ksid Vector form of the equation of plane. ;,hl iïlrKfha ffoYsl wdldrh' A P

Vector form of the equation of plane. cartesian form ldÜishdkq wdldrh" fuys

ksoiqk ( 2, 3, 1 ),laIHh yryd g,usnlj we;s ;,fha iïlrKh fidhkak" ;,fha iïlrKh f,i ;,fha iïlrKh,efns' ksoiqk ( -2, 1, 1 ), ( 3, 0, 2 ) iy ( 2, 4, 5 ),laIHh yryd jQ ;,fha iïlrKh fidhkak'

ksoiqk ( -2, 1, 1 ), ( 3, 0, 2 ) iy ( 2, 4, 5 ),laIHh yryd jQ ;,fha iïlrKh fidhkak' A ( -2, 1, 1 ) B ( 3, 0, 2 ) C ( 2, 4, 5 ) AB = AC = ;,hg,usnl ffoYslhla

fus ksid ;,fha iïlrKh fyda ksoiqk P(1,0,0), Q(0,1,0) iy R(0,1,1),laIHh yryd jQ ;,fha iïlrKh fidhkak' Find the Equation of a plane containing the points P(1,0,0), Q(0,1,0) and R(0,1,1).

E.g.Find the Equation of a plane containing the points P(1,0,0), Q(0,1,0) and R(0,1,1). P Q R A A (x, y, z) variable point vectors, and are coplanar.

;, foll fPaokh' fok,o ;, folla iudka;r fyda fPaokh fyda fjS' iy u.ska fokq,nk ;, iudka;r kus fPaokh fjs kus ;, foflys fPaokh ir, f¾Ldjla fjS'

fPaok f¾Ldj ffoYslh fPaokh f¾Ldjg iudka;r fjS' fuS ksid fPaokh f¾Ldj u;,laIHhla fidhd.;a úg fPaokh f¾Ldfjs iïlrKh,nd.; yel"

ksoiqk iy u.ska fokq,nk ;, fPaokh fjS oehs fidhkak' tksid ;, fPaokh fjS' u.ska fokq,nk ;,h i|yd i|yd ;,h u; fmd¥,laIHhla f,i.ksuq' túg

tksid ( 2, 1, 0 ),laIHh ;,h u; fjS" ;,fha iïlrKh