MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER

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1 MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER I One M Queston Fnd the unt veto n the deton of Let ˆ ˆ 9 Let & If Ae the vetos & equl? But vetos e not equl sne the oespondng omponents e dstnt e detons e dffeent Fnd the vlues of x & y so tht vetos nd x e equl Gven x y x y x y Fnd the sl o dot podut of vetos & ( (

2 unt veto s unt veto s Show tht II Two Ms Questons: Fnd the veto pllel to the veto nd hs mgntude unts ˆ Let veto Reqd ˆ Now mgntude unts hvng to the vetos pllel e Let the veto of the deton tos nd deton osnes Fnd ˆ ˆ QIf vetos e pllel then unt veto long & pllel vetos e sme

3 Hee deton tos e omponents of deton osnes e omponents of â e e ( Show the vetos nd 8 e ollne 8 ( One veto n e expessed ntems of nothe & e ollne Fnd x f fo unt veto Gven ( x ( x x x x x ( x ( x Fnd f two vetos nd e suh tht Gven nd w t ( 8 9 nd

4 t w os os of vlues fo ll os os nd pove tht Fo ny two vetos { } Q popetes Fom pevous n equlty Tngle Fo ny two vetos nd pove tht Evlute 8 { } { } { } { } nd fnd & If 9

5 nd Equtng offents Gven f & Fnd µ µ µ µ µ µ µ µ µ o o o o ( ( ( ( ( ( Consde LHS( ( ( Show tht ( o o podut Sl tple nd vetos the sl tple podut of Fnd vetos e opln Gven tht e opln & the vetos f Fnd

6 III Thee Ms Questons: Consde the ponts P nd Q wth poston vetosop nd OQ Fnd the poston veto of pont R whh dvdes lne onng the ponts P nd Q n the to :ntenlly nd extenlly espetvely Soluton gven OP OQ m : n : Intenlly moq nop OR m n ( ( OR moq nop extenlly OR m n ( ( OR Fnd the veto onng the ponts P( nd Q ( nd lso deton osnes of PQ

7 Gven P then OP OQ PQ OQ OP PQ ( Q ( Let e unt vetos long xes PQ ( ( ( 9 PQ ^ PQ PQ PQ ^ / PQ / deton osnes e Show tht ponts A( B( & C( e ollne OR Show tht the ponts wth poston vetos nd e ollne OA AB BC AC AB ( OB ( OC ( OB OA ( ( OC OB ( ( ( OC OA ( ( ( 8 8 BC AC AC Hee AB BC Collne ondton s stsfed A AC B C e ollne

8 Fnd the ngle etween the vetos Let ( ( ( Let e ngle etween & then os os os & Show tht the vetos e mutully pependul Let ( ( ( onsde 9 { 8 } { } 9 9 s pependul to ly /// We n show tht & e mutully pependul vetos ( ( nd ( ( ( {( ( } 8

9 9 pependul to s 8 8 onsde 8 e pependul & vetos then show tht the If { } { } 8 8 s Gven then fnd pependul to s nd suh tht nd If 8 /// then fnd e unt vetos suh tht If 9 Addng o o ly Q

10 Fnd veto nd unt veto pependul to eh of the veto nd whee & Let e the veto pependul to ĉ 8 ( ( ( 8 8 ( 8 Let ĉ e the unt veto pependul to then ĉ ( nd ( then ( ( ( nd

11 o o os os os os os os os os os os os os then os Gven Let then wth mde y the ngles e Let nd hene omponents of wth then fnd nd nd ute ngle wth wth ngles mes unt veto If ± γ π β π α γ π β π α γ β α π π

12 Fnd the e of tngle wth vetes A A B C ( ( ( OA AB OB OA AB AC OC OA AC OB OC B( C( AB AC AB AC ( ( ( ( ( Ae of tngle ABC AB AC 9 sq unts

13 Fnd the e of pllelogm whose deent sdes e nd Also fnd unt veto pllel to ts dgonl AC AC Unt veto long AC { } { } { } Ae of pllelogm ABCD sq unts Dgonl AC ( ^ AC 9 AC AC 9 Unt veto pllel to AC Unt veto long AC Show tht the ponts & Gven A A B espetvely fom the vetes of C wth poston vetos ( B ( C ( Gven OA ght ngled tngle

14 stsfed lw s O CA BC AB CA BC AB OC OA CA BC OB OC BC AB OA OB AB OC OB le tngle ght ngled fom e stfed Pythgous theoem s tht We n see BC AC AB BC AC AB CA CA BC BC AB AB fom le

15 [ ] [ ] { } { } { } [ ] LHS tht Pove { } { } { } opln e vetos onsde Let e opln nd vetos Show tht the Sl Tple Podut of vetos Popetes [ ] [ ] e stsfed popetes detemnent ll detemnent sne sl tple podut s Then o denoted y s The sl tple podut of vetos lled sl tple podut of s then e non zeo vetos If o

16 ( s sl ( ( ( [ ] [ ] [ ] ( ( [ ] [ ] ( ( [ ] [ ] et Dot nd oss n e ntehnged ( ( ( ( et Copln veto: The vetos e sd to e opln f they le on sme plne o pllel plnes ( The ondton fo the vetos to e opln s Polems: Fnd the sl tple podut of vetos nd Sl tple podut (

17 { } { } { } opln e vetos onsde [ ] [ ] Pove tht { } { } { } [ ] LHS opln e nd vetos tht the Show Let opln e & vetos the f Fnd

18 8 vetos e opln Gven tht

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