TOPIC 10 THREE DIMENSIONAL GEOMETRY
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1 TOPIC THREE DIMENSIONAL GEOMETRY SCHEMATIC DIAGRAM Topi Conept Degree of importne Three Dimensionl Geometr (i Diretion Rtios n Diretion Cosines (iicrtesin n Vetor eqution of line in spe & onversion of one into nother form Refrene NCERT Tet Book Eition 7 * E No Pg -466 E No 5 Pg 467 E No 4 Pg - 48 ** E No 8 Pg -47 Q N Pg 477 QN 9 Pg 478 (iiico-plner n skew lines * E No 9 Pg -496 (iv Shortest tne etween two lines (v Crtesin n Vetor eqution of plne in spe & onversion of one into nother form (vi Angle Between (iv Two lines (v Two plnes (vi Line & plne (vii Dtne of point from plne (viiidtne mesures prllel to plne n prllel to line (ieqution of plne through the intersetion of two plnes ( Foot of perpeniulr n imge with respet to line n plne *** E No Pg -476 Q N Pg 478 ** E No 7 Pg -48 E No 8 Pg 484 E No 9 Pg 485 E No 7 Pg 495 Q N. 9 - Pg 499 * * ** E No 9 Pg -47 Q N. - Pg 478 E No 6 Pg 494 Q N. - Pg 494 E No 5 Pg - 49 ** Q No 8 Pg -499 Q No 4 Pg 494 ** *** Q No Pg -49 ** E. N 6 Pg 48 SOME IMPORTANT RESULTS/CONCEPTS ** Diretion os ines n iretion rtios : If line mkes ngles n with n es respetivel theos os n os re theiretion os ines enote l m n respetivel n l m n
2 Anthreenumers proportionl to iretion os ines re iretion rtios enote l m n l m n * Diretion rtios of line segment joining P * Angle etween twolines whoseiretion os ines re l m n os l l m m * For prllellines n n for perpeniulr lines ** STRAIGHTLINE : * * then Shortest tne n Q n Eqution of line (Vetor form. Eqution of line pssing through point or l l * Eqution of line pssin g through two point * Eqution of line pssin g through point * Eqution of line pssin g through point m m n l m n n n n * Shortest tne etween two skew lines : if lines re r ; ; m e tken s given n in the iretion of r r Eqution of line pssing through two points & n in the iretion of r ** PLANE: * Eqution of plne * Eqution of plnein interept form * Eqution of plnein norml form l length of perpeniulr form origin totheplne = where & re iretion rtios of *Eqution of plne pssing through point ( m n with iretion os ines : n prllel totheline: wherereint ereptson thees p wherelmn re iretion os ines of norml totheplne norml to the plne p
3 or prrll totheline n perpeniulr to the plne ( plne pssing through two points( *Eqution of n ( ( plne pssing through three points( *Eqution of ( n pssing through the point plne ontning theline * Eqution of n tothelines prrll or plnes n perpeniulr to the ( plne pssing through the point * Eqution of ` plnes prllel * Dtne etween two to the plne tn e from thepoint *Perpeniulr ( ( twoplnes se tion of er plne pssing through the int * Eqution of plne ommon of n eqution if re oplner n : * Conition for oplner lines ASSIGNMENTS (idiretion Rtios n Diretion Cosines
4 LEVEL-I. Write the iretion-osines of the line joining the points ( n ( [CBSE ].Fin the iretion osines of the line pssing through the following points (-4-5 (..Write the iretion osines of line equll inline to the three oorinte es LEVEL-II.Write the iretion osines of line prllel to the line = =..Write the iretion rtios of line prllel to the line = =.. If the eqution of line AB Fin the iretion osine. 4. Fin the iretion osines of line pssing through origin n ling in the first otnt mking equl ngles with the three oorinte. (ii Crtesin n Vetor eqution of line in spe & onversion of one into nother form LEVEL-I.Write the vetor eqution of the line = =. [CBSE ]. Write the eqution of line prllel to the line = = n pssing through the point(..epress the eqution of the plne = ( + + λ( + + in the Crtesin form. 4.Epress the eqution of the plne. ( = in the Crtesin form. (iii Co-plner n skew lines LEVEL-II.Fin whether the lines = ( λ( + n = ( - + µ( + - interset or not. If interseting fin their point of intersetion..show tht the four points (-- (45 (94 n (-444 re oplnr. Also fin the eqution of the plne ontining them. 4.Show tht the lines n interset. Fin their point of 4 5 intersetion. LEVEL-III. Show tht the lines = = n = = re oplnr. Also fin the eqution of the plne.. The points A(45 B(4 n C(- re three verties of prllelogrm ABCD. Fin
5 the vetor eqution of the sies AB n BC n lso fin the oorintes.fin the equtions of the line whih intersets the lines = = n = = n psses through the point (. 4. Show tht The four points ( (4 5 ( 9 4 n ( re oplnr n fin the eqution of the ommon plne. (iv Shortest tne etween two lines LEVEL-II. Fin the shortest tne etween the lines l n l given the following: ( l : = = l : = = ( = ( λ( + =(4 + µ + (5 +µ + (6 + µ.. Show tht the lines n 4 intersetion... Fin the shortest tne etween the lines 4 5 interset. Fin their point of r = ( î + ĵ + ( î ĵ + kˆ n r = ( î + ĵ kˆ + (4 î ĵ + kˆ 4.Fin the shortest tne etween the lines = ( ( ( n = ( ( ( [CBSE ] 5. Fin the tne etween the prllel plnes + = -4 n + + =. 6. Fin the vetor eqution of the line prllel to the line = = n pssing through (-4. Also fin the tne etween these two lines. (v Crtesin n Vetor eqution of plne in spe & onversion of one into nother form LEVEL I.Fin the eqution of plne pssing through the origin n perpeniulr to -.Fin the eqution of plne with interepts 4 on the respetivel.. 4. ( ( ( ( (
6 LEVEL II. Fin the vetor n rtesin equtions of the plne whih psses through the point (5 4 n perpeniulr to the line with iretion rtios.. Fin the vetor eqution of plne whih t tne of 7 units from the origin n norml to the vetor iˆ + 5 ˆj - 6 kˆ..fin the vetor n rtesin equtions of the plnes tht psses through the point ( n the norml to the plne iˆ + ˆj - kˆ. (vi Angle Between(iTwo lines (iitwo plnes (iiiline & plne LEVEL-I. Fin the ngle etween the lines whose iretion rtios re ( n ( 4..Fin the ngle etween line = = n the plne =..Fin the vlue of λ suh tht the line = = 7. λ = perpeniulr to the plne.fin the ngle etween the plnes whose vetor equtions re r ( iˆ + ˆj - kˆ = 5 n r ( iˆ - ˆj + 5 kˆ = 5.Fin the ngle etween the line = = n the plne + =. LEVEL-II.Fin the vlue of p suh tht the lines n re perpeniulr to eh other.. A line mkes ngles α β γ δ with the four igonls of ue Prove tht os α + os β + os γ + os δ =. (vii Dtne of point from plne LEVELI.Write the tne of plne + + = from the origins..fin the point through whih the line = = 4 psses. ( ( î ĵ kˆ 4. Fin the tne of the following plne from origin: + + = 5.Fin the tne of the point ( from -
7 ..Fin the points on the line LEVEL II = = t tne of 5 units from the point P(..Fin the tne of the point (45 from the plne + + = mesure prllel to the line = =.. Fin the tne etween the point P(6 5 9 n the plne etermine the points A ( B (5 4 n C( ( î ĵ kˆ λ. î ĵ kˆ / ( î ĵ kˆ - LEVEL III.Fin the oorintes of the foot of the perpeniulr n the perpeniulr tne of the point (4 from the plne + + =. Fin lso the imge of the point in the plne..fin the tne of the point P(659 from the plne etermine the points A(- B(54 n C(--6..Fin the eqution of the plne ontining the lines = + + λ( + - n = + + µ(- + -.Fin the tne of th plne from origin n lso from the point (. (viii Eqution of plne through the intersetion of two plnes LEVELII.Fin the eqution of plne pssing through the point ( n perpeniulr to the line joining the points (4 n (5. Also fin the perpeniulr tne of the plne from the origin..fin the eqution of the plne whih perpeniulr to the plne = n whih ontins the line of intersetion of the plnes = n =..Fin the eqution of the plne tht ontins the point (- n perpeniulr to eh of the plnes + - = 5 n + = 8. LEVEL-III.Fin the eqution of the plne pssing through the point ( n ontining the line = ( λ( Also show tht the plne ontins the line = ( λ( Fin the eqution of the plne pssing through the point ( n perpeniulr to the plnes = n + 4 =..Fin the Crtesin eqution of the plne pssing through the points A( n B(- n prllel to the line = = ` 4. Fin the eqution of the perpeniulr rwn from the point P(4- to the line = =.
8 (ifoot of perpeniulr n imge with respet to line n plne LEVEL II. Fin the oorintes of the point where the line through (-4-5 n (- rosses the plne etermine points A( B( n C(-6.. Fin the foot of the perpeniulr from P( on the line = =. Also otin the eqution of the plne ontining the line n the point (..Prove tht the imge of the point (- in the plne + 4 = lies on the plne =. LEVEL-III.Fin the foot of perpeniulr rwn from the point A( to the joint of the points B(4 7 n C( 5.. Fin the imge of the point ( in the line.. The foot of the perpeniulr from the origin to the plne ( 4. Fin the eqution of the plne 4. Fin the oorintes of the foot of the perpeniulr n the perpeniulr tne of the point P( from the plne ++=. Fin lso the imge of the point in the plne. Questions for self evlution. Fin the eqution of the plne pssing through the point ( n perpeniulr to the plnes = n + 4 =.. Fin the vetor eqution of line joining the points with position vetors î ĵ kˆ n prllel to the line joining the points with position vetors î ĵ + 4 kˆ n î + ĵ + kˆ. Also fin the rtesin equivlent of th eqution.. Fin the foot of perpeniulr rwn from the point A( to the joint of the points B(4 7 n C( Fin the shortest tne etween the lines r = ( î + ĵ + ( î ĵ + kˆ n r = ( î + ĵ kˆ + (4 î ĵ + kˆ 5.Fin the imge of the point ( in the line. 6. Show tht the four points ( (4 5 ( 9 4 n ( re oplnr n fin the eqution of the ommon plne. 7. The foot of the perpeniulr from the origin to the plne ( 4. Fin the eqution of the plne Show tht the lines n interset. Fin their point of 4 5 intersetion. 9. A line mkes ngles α β γ δ with the four igonls of ue Prove tht os α + os β + os γ + os δ =.
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