CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER ......Find the vector equation of line passing through...

CENGAGE/GTEWANIMATHSSOLUTIONS

CHAPTER THREE DIMENSIONAL GEOMETRY ||VECTORSAND3DGEOMETRY

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1

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_DirectionCosinesAndDirectionRatios

If aretheangleswhichadirectedlinemakeswiththepositivedirectionsoftheco-ordinatesaxes,thenfindthevalueof

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2


Aline throughorigin isinclinedat respectivley.Thenfindtheangleatwhichitisinclinedto


3


isatriangleandA=(2,3,5),B=(-1,3,2)andC= Ifthemedianthroughisequallyinclinedtotheaxes,thenfindthevalueof and


4


A line passes through the points Find te directioncosinesoffthelineifthelinemakesanacuteanglewiththepositivedirectionofthex-axis.


5


Find the ratio in which the plane divides the join of the points


α, β, andγsin2α + sin2 β + sin2 γ.

OP O 300and450 → OXandOY ,OZ.

ABC (λ, 5, μ).A λ μ

(6, − 7, − 1)and(2, − 3, 1).

y − z( − 2, 4, 7)and(3, − 5, 8).

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6


If

arethreecollinearpoints,thenfindtheratioinwhichpoint divides


7


Ifthesumofthesquaresofthedistanceofapointfromthethreecoordinateaxesis36,thenfinditsdistancefromtheorigin.


8


Alinemakesangles withthediagonalsofacube.Showthat


9


Findtheanglebetweenthelinewhosedirectioncosinesaregivenby

A(3, 2, − 4),

B(5, 4, − 6)andC(9, 8,

− 10)C AB.

α, β, γandδcos2 α + cos2 β + cos2 γ

+ cos2 δ = 4/3.

l +m + n = 0and2l2 + 2m2

− n2 − 0.

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10


Amirror and a source of light are situated at the origin and at a point onrespectively.A rayof light from the source strikes themirror and is reflected. If thedirection ratiosof thenormal to theplaneare then find the of thereflectedray.


11

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_Equation Of Straigth Line Passing Through A Given Point AndParallelToAGivenVector

The Cartesian equation of a line is . Find the vector

equationoftheline.


12

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_Equation Of Straigth Line Passing Through A Given Point AndParallelToAGivenVector

The Cartesian equations of a line are Find itsdirectionratiosandalsofindavectorequationoftheline.


13

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_EquationOfLinePassingThroughTwoGivenPoint

A line passes through the point with position vector and is in thedirection of Find the equations of the line in vector and Cartesianforms.

O OX,

1, − 1, 1, DCs

= =x − 32

y + 1−2

z − 35

6x − 2 = 3y + 1 = 2z − 2.

2 î − 3 ĵ + 4k̂3î + 4 ĵ − 5k̂.

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14


Findthevectorequationoflinepassingthrough AlsofinditsCartesianequations.


15


Find Cartesian and vector equation of the line which passes through the point

andparalleltothelinegivenby .


16


Find theequationofa linewhichpasses through thepoint andwhich hasequalinterceptsontheaxes.


17


Findthepointswhereline intersects planes.



A(3, 4 − 7)andB(1, − 1, 6).

( − 2, 4, − 5) = =x+ 33

y − 45

z + 86

(2, 3, 4)

= =x − 12

y + 2−1

z

1xy, yzandzx

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18Findtheequationofline

insymmetricform.Findthedirectionoftheline.


19


Find the vector equation of line passing through the point andperpendiculartothetwolines:


20


If

aretwolines,thenfindtheequationofacuteanglebisectoroftwolines.


21


Find the equation of the line drawn through point (1, 0, 2) to meet the line

atrightangles.



Line is parallel to vector and passes through a point

x + y − z − 3 = 0 = 2x+ 3y + z + 4

(1, 2, − 4)

=

= and

= =

x − 83

y + 19−16

z − 107

x − 153

y − 298

z − 5−5

→r = ( î + 2ĵ + 3k̂)+ λ(î − ĵ + k̂)and→r= ( î + 2ĵ + 3k̂)+ μ( î + ĵ + k̂)

= =x + 13

y − 2−2

z1−1

L1→α = − 3 î + 2ĵ + 4k̂

→

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22 and line is parallel vector and point

Nowaline paralleltoavector intersects the lines

atpoints respectively,thenfind


23


Find thecoordinatesofapointon the atgadistance

fromthepoint


24

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_AngleBetweenTwoLines

Findtheangelbetweenthefollowingpairoflines:



A(7, 6, 2) L2→β = 2 î + ĵ + 3k̂ B(5, 3, 4).

L3→r = 2 î − 2ĵ − k̂ L1andL2

CandD, ∣∣∣→C D

∣∣∣.

= = zx − 12

y + 1−3

4√14

(1, − 1, 0).

→r = 2 î − 5ĵ + k̂

+ λ(3 î + 2ĵ + 6k̂)and→r= 7 î − 6k̂ + μ(î + 2 ĵ+ 2k̂)= = and

= =

x

2y

2z

1x− 54

y − 21

z− 38

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25

Findthevalues sothatline

areatrightangles.


26


Findtheacuteanglebetweenthelines

aretherootsofthecubicequation


27


Fidtheconditioniflines

areperpendicular.


28

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_PerpendicularDistanceOfAPointFromALine

Find thecoordinatesof the footof theperpendiculardrawn frompoint tothejoinofpoints


29


Find the length of the perpendicular drawn from point to line


p

=

= and

= =

1 − x3

7y − 142p

z − 32

7 − 7x3p

y − 51

6 − z5

= = and

= =

= wherel > m > n,

andl,m, n

x − 1l

y + 1m

1n

x + 1m

y − 3n

z − 1l

x3 + x2 − 4x = 4.

x = ay + b, z = cy + dandx= a ′ y + b ′ , z = c ′ y + d'

A(1, 0, 3)B(4, 7, 1)andC(3, 5, 3).

(2, 3, 4)

= = .4 − x2

y

61 − z3

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30

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_ShortestDistanceBetweenTwoLines

Findtheshortestdistancebetweenthelines

.


31


Determinewhetherthefollowingpairoflinesintersectornot.(1)

(2)



Findtheshortestdistancebetweenlines

=

= and

= =

x − 12

y − 23

z − 34

x − 23

y − 44

z− 55

→r = î − 5 ĵ + λ(2î + k̂);→r = 2 î − ĵ

+ μ( î + ĵ − k̂)→r = î + ĵ − k̂

+ λ(3 î − ĵ); →r = 4 î − k̂+ μ(2î + 3k̂)

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32


33


Ifthestraightlines

withparamerters respectivley,arecoplanar,thenfind


34

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_Plane

Findtheequationofalinewhichpassesthroughthepoint andintersectsthelines


35


Findtheequationofplanewhichisatadistance fromtheoriginandisnormal

tovector


→r = ( î + 2ĵ + k̂)+ λ(2 î + ĵ + 2k̂)and→r= 2 î − ĵ − k̂ + μ(2î + ĵ+ 2k̂).

x = − 1 + s, y = 3 − λs, z

= 1 + λsandx = , y = 1

+ t, z = 2 − t,

t

2

sandt, λ.

(1, 1, 1)

=

= and

= = .

x − 12

y − 23

z − 34

x + 21

y − 32

z+ 14

4

√142î + ĵ − 3k̂.

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36


Findtheunitvectorperpendiculartotheplane


37


Findthevectorequationofa linepassing through andperpendiculartotheplane


38


Findtheequationoftheplanepassingthroughthepoint having asthedirectionratioisofthenormaltotheplane.


39


Thefootoftheperpendiculardrawnfromtheorigintoaplaneis Findtheequationoftheplane.orIf istheoriginandthecoordinatesof isthenfindtheequationoftheplanepassingthrough andperpendicularto


40


Findtheequationoftheplanesuchthatimageofpoint initis


→r

.

2î + ĵ + 2k̂ = 5.

3î − 5 ĵ + 7k̂3x − 4y + 5z = 8.

(2, 3, 1) (5, 3, 2)

(1, 2, − 3).O P (1, 2, − 3),

P OP .

(1, 2, 3) ( − 1, 0, 1).

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41


Find the equation of the plane passing through Alsofindaunitvectorperpendiculartothisplane.


42


Showthatthelineofintersectionoftheplanes

isequallyinclinedto Alsofindtheangleitmakeswith


43


FindthevectorequationofthefollowingplanesinCartesianform:



Provethattheplane containstheline

A(2, 2, − 1), B(3, 4,2)andC(7, 0, 6).

→r

.

î + 2 ĵ + 3k̂ = 0and→r

= (3î + 2 ĵ + k̂) = 0iandk. j.

→r = î − ĵ + λ(î + ĵ + k̂)+ μ( î − 2ĵ + 3k̂).

→r = (î + 2 ĵ − k̂) = 3

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44


45


Findtheequationoftheplanewhichisparalleltothelines

andispassingthroughthepoint ).


46


If a plane meets the equations axes at such that the centroid of thetriangleis thenfindtheequationoftheplane.


47


Findtheequationoftheplanepassingthrough whichisparallel tothe

plane


→r = î + ĵ + λ(2 î + ĵ+ 4k̂).

→r = î + ĵ + λ(2 î + ĵ+ 4k̂)and ==

x+ 1−3

y − 32

z + 21

(0, 1, − 1

A,BandC(1, 2, 4),

(3, 4, − 1),→r

.

2î − 3 ĵ + 5k̂+ 7 = 0.

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48

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_AngleBetweenTwoPlanes

Findtheangelbetweentheplanes


49

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_AngleBetweenTwoPlanes

Showthat

areperpendicularto planes,respectively.


50

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_LineOfIntersectionOfTwoPlanes

Reducetheequationofline

insymmetricalform.OrFindthelineofintersectionofplanes



Findtheanglebetweenthelines

2x+ y − 2x + 3

= 0and→r.

6 î + 3ĵ + 2k̂ = 5.

ax + by + r = 0, by + cz+ p = 0andcz+ ax + q= 0

x− y, y − zandz − x

x − y + 2z = 5adn3x + y+ z = 6

x − y + 2z = 5and3x + y+ z = 6.

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51


52


Iftheline intersecttheline

thenfindthevalueof

aretheanglesofatriangle.


53


Find the point of intersection of line passing through and the intersectionlines

withthe plane.



x − 3y − 4 = 0, 4y − z+ 5= 0andx + 3y − 11 = 0, 2y= z + 6 = 0.

x = y = zs ∈ A

.x + s ∈ B

.y + s ∈ C

.z

= 2d2, s ∈ 2A.x + s ∈ 2B

.y

+ s ∈ 2C.z = d2,

whereA,

B, C

sinA2

.sinB2

.sinC2

(0, 0, 1)

x + 2u + z = 1, − x + y− 2zandx + y = 2, x+ z= 2

xy

4 − 3 + 7 = 0

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54 Ahorizontalplane isgiven.Findalineofgreatestslopepassesthroughthepoint intheplane


55


Find the equation of the plane passing through the points andandperpendiculartotheplane


56


Findtheequationoftheplanecontainingline andpoint


57


Findthedistanceofthepoint fromtheline

measuredparalleltotheplane


58


Findthedistanceofthepoint fromtheplane measured

paralleltotheline


59


Find the equation of the projection of the line on the

plane


4x − 3y + 7z = 0(2, 1, 1) 2x+ y − 5z = 0.

( − 1, 1, 1)(1, − 1, 1) x + 2y + 2z = 5.

= =x + 1−3

y − 32

z + 21

(0, 7, − 7).

P(3, 8, 2)

(x − 1) = (y − 3)

= (z − 2)

12

14

13

3x + 2y − 2z + 15 = 0.

(1, 0, − 3) x− y − z = 9

= = .x − 22

y + 22

z − 6−6

= =x− 12

y + 1−1

z − 34

x+ 2y + z = 9.

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60

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_AngleBetweenALineAndAPlane

Findtheanglebetweentheline

andtheplane


61


Find the vector equation of the line passing through and parallel to theplanes


62

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_EquationOfAPlanePassingThroughTheLineOfIntersectionOfTwoPlanes

Findtheequationtheplanewhichcontainthelineofintersectionoftheplanes

andwhichisperpendiculartotheplane .


→r = î + 2 ĵ − k̂

+ λ(î − ĵ + k̂)→r

.

2 î − ĵ + k̂ = 4.

(1, 2, 3)

→r

.

î − ĵ + 2k̂and→r

.

3 î + ĵ + k̂= 6.

→r

.

î + 2 ĵ + 3k̂− 4

= 0and→r.

2 î + ĵ − k̂ + 5= 0

→r (5 î + 3ĵ − 6k̂) + 8 = 0

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63


Findtheequationofaplanecontainingthelineofintersectionoftheplanes

passingthrough .


64


Theplane is rotated throughanangle about its lineof intersectionwith the plane Show that he equation to the plane in the new position is


65

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_DistanceOfAPointFromAPlane

Findthelengthandthefootoftheperpendicularfromthepoint totheplane


66

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_DistanceBetweenParallelPlanes

Findthelocusofapoint,thesumofsquaresofwhosedistancefromtheplanes

.


x + y + z − 6 = 0and2x+ 3y + 4z + 5 = 0

(1, 1, 1)

ax+ by = 0 αz = 0.

aby ± z√a2 + b2andα = 0.

(7, 14, 5)2x+ 4y − z = 2.

x − z = 0, x− 2y + z= 0andx + y + z = 0is36

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67


A ray of light passing through the point , strikews the plane and on reflection passes through point Find the

coordinatesopoint


68


Findthedistancebetweentheparallelplanes


69


Find the image of the line in the plane


70

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_EquationOfAPlaneBisectingTheAngleBetweenTwoPlanes

Findtheequationsofthebisectorsoftheanglesbetweentheplanes

andspecifytheplanewhichbisectstheacuteangleandtheplanewhichbisectstheobtuseangle.


71

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_TwoSidesOfAPlane

Findtheequationofaspherewhosecentreis radiusis


A(1, 2, 3)xy + z = 12atB C(3, 5, 9).

B.

x + 2y − 2z + 1 = 0and2x+ 4y − 4z + 5 = 0.

= =x− 19

y − 2−1

z + 3−3

3x− 3y + 10z − 26 = 0.

2x− y + 2z + 3 = 0and3x− 2y + 6z + 8 = 0

(3, 1, 2) 5.

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72


Findtheequationofthespherepassingthrough


73


Find theequationof thespherewhichhascentreat theoriginand touches the line


74


Findtheequationofthespherewhichpassesthroughandwhosecentreliesontheplane


75


Findtheequationofaspherewhichpassesthroughandhasradiusassmallaspossible.



Findthelocusofappointwhichmovessuchthatthesumofthesquaresofitsdistancefromthepoints

(0, 0, 0), (1, 0, 0),

( − , 1, 0)and(0, 0, 1).

2(x + 1) = 2 − y = z + 3.

(10, 0), (0, 1, 0)and(0, 0, 1)3x − y + z = 2.

(1, 0, 0)(0, 1, 0)and(0, 0, 1),

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76


77


Find the equation of the sphere described on the joint of points havingpositionvectors

respectively,asthediameter.Findthecenterandtheradiusofthesphere.


78


Findtheradiusofthecircularsectioninwhichthesphere iscutbytheplane


79


Showthattheplane touchesthesphere

A(1, 2, 3),

B(2, − 3, 5)andC(0, 7,

4)is120.

AandB

2 î + 6ĵ − 7k̂and − 2 î + 4ĵ

− 3k̂,

∣∣→r ∣∣ = 5

→r

.

î + ĵ + k̂ = 3√3.

2x − 2y + z+ 12 = 0x2 + y2 + z2 − 2x − 4 + 2z− 3 = 0.

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80


Avariableplanepassesthroughafixedpoint andcutsthecoordinateaxesatpoints Show that eh locus of the centre of the sphere


81


A sphere of constant radius passes through the origin and meets the axes at Prove that the centroid of triangle lies on the sphere


82


Ifthex-coordinateofapoint onthejoinof

thenfindits coordinate.


83


Findthedistanceofthepoint fromthex-axis.


(a, b, c)A, B, andC.

OABCis + + = 2.a

x

b

y

c

z

k,A, BandC . ABC9(x2 + y2 + z2) = 4k2 .

PQ(22, 1)andR(5, 1,

− 2)is4,z −

P(a, b, c)

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84


If isavectorofmagnitude21andhasdirectionratios thenfind


85


If is a point on the line segment joining such

thattheprojectionsof ontheaxesare13/5,19/5and26/5,respectively,thenfindtheratioinwhich divides


86


If is the origin, with direction ratios then find thecoordinatesof


87


If a line makes angles with threew-dimensional coordinate axes,respectively,thenfindthevalueof


88


Alinemakesangles withthecoordinateaxes.If thenfind

→r 2, − 3and6, →r .

P (x, y, z) Q(2, 2, 4)andR(3, 5, 6)→OP

P QR.

O OP = 3 −1, 2, and − 2,P .

α, βandγcos 2α + cos 2β + cos2γ.

α, βandγ α + β = 900,γ.

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89


The line joining thepoints is parallel to the linewhosedirectionratiosare Findthevaluesof


90


A parallelepiped is formed by planes drawn through the pointsparallel to thecoordinateplanes.Find the lengthofedges

anddiagonaloftheparallelepiped.


91

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_RegularTetrahedron

Findtheanglebetweenanytwodiagonalsofacube.


92

CENGAGE_MATHS_VECTORS AND 3D GEOMETRY_THREE DIMENSIONALGEOMETRY_Spheres

Direction ratios of two lines are Then the lines are______.



Find the angle between the lines whose direction cosines are connected by the

( − 2, 1, − 8)and(a, b, c)6, 2, and3. a, b and c

P(6, 8, 10)and(3, 4, 8)

a, b, cand1/ bc, 1/ ca, 1 /ab.

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93 relations


94


Findthepointwherelinewhichpassesthroughpoint andisparalleltoline

meetsthexy-plane.


95


Findtheequationofthelinepassingthroughthepoints



Findtheequationofthelinepassingthroughthepoint andperpendiculartothelines

l +m + n = 0and2lm+ 2nl− mn = 0.

(1, 2, 3)→r = î + ĵ + 2k̂

+ λ(î − 2 ĵ + 3k̂)

(1, 2, 3)and( − 1, 0, 4).

( − 1, 2, 3)

y − 1

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96


97


Findtheequationofthelinepassingthroughtheintersectionof

andalsothroughthepoint


98


Thestraightline is(a)Paralleltox-axis(b)Paralleltothe

y-axis(c)Paralleltothez-axis(d)Perpendiculartothez-axis


99


Findtheangelbetweenthelines


100


Ifthelines

=

= and

= = .

x

2y − 1−3

z + 2−2

x + 3−1

y + 32

z− 13

=

= and

= = z.

x − 12

y − 2−3

z − 34

x − 45

y − 12

(2, 1, − 2).

= =x − 33

y − 21

z − 10

2x = 3y = − zand6x =− y = − 4z.

x − 1 y − 2

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areatrightangle,thenfindthevalueof


101


Theequationsofmotionofarocketare

wheretime isgiveninseconds,andthecoordinatesofamovingpointsinkilometers.What is thepathof the rocket?Atwhatdistancewillbe the rocket from thestartingpoint in


102


Find the length of the perpendicular drawn from the point to the line

wher isaparameter.

=

= and

= =

x − 1−3

y − 22k

z − 3−2

x − 13k

y − 51

z− 6−5

k.

x = 2t, y = − 4tandz= 4t,

t

O(0, 0, 0) 10s?

(5, 4, − 1)→r = î + λ(2î + 9ĵ + 5k̂), λ

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103


Findtheimageofpoint intheline


104


Findtheshortestdistancebetweenthelines


105


Findtheshortestdistancebetweenthez-axisandtheline,


106


Ifthelines

intersect,thenfindthevalueof


107


Findtheplaneoftheintersectionof

(1, 2, 3) = = .x − 63

y − 72

z − 7−2

→r = (1 − λ)î + (λ − 2) ĵ

+ (3 − 2λ)k̂and→r

= (μ + 1) î + (2μ + 1)k̂.

x + y + 2z − 3 = 0, 2x + 3y+ 4z − 4 = 0.

=

= and

= =

x − 12

y + 13

z − 14

x − 31

y − k2

z

1k.

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108


Find the radius of the circular section of the sphere by the plane


109


A point is such that where are the point irrespectivley. Find the equation to the locus of the

point andverifythatthelocusisasphere.


110


Theextremitiesofadiameterofaspherelieonthepositivey-andpositivez-axesatdistance2and4, respectively.Show that thespherepasses through theoriginandfindtheradiusofthesphere.



x2 + y2 + z2 + 2x + 2y + 2

= 0and4x2 + 4y2 + 4z2

+ 4x + 4y + 4z − 1 = 0.

∣∣→r ∣∣ = 5

→r

.

î + 2 ĵ − k̂ = 4√3.

P(x, y, z) 3PA = 2PB, AandB(1, 3, 4)and(1, − 2, − 1),

P

( , , )

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111

Aplanepassesthroughafixedpoint Showthatthelocusofthefootoftheperpendiculartoitfromtheoriginisthesphere


112


Find the angle between the line and the plane


113


Find the distance between the line and the plane


114


Findthedistanceofthepoint fromthepointofintersectionofthe

line andplane



(a, b, c).

x2 + y2 + z2 − ax − by − cz= 0.

= =x − 13

y − 12

z − 14

2x+ y − 3z + 4 = 0.

= =x + 1−3

y − 32

z − 21

x + y + z + 3 = 0.

( − 1, − 5, − 10)

= =x − 23

y + 14

z− 212

x − y + z = 5.

( − 1, 3, 2)

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115

Find the equation of the plane passing through the point andperpendiculartoeachoftheplanes


116


Findtheequationoftheplanecontainingthelines


117


Find theequationof the planepassing through thepoints

andparalleltotheline


118


Find the equation of the plane passing through the straight line

andperpendiculartotheplane


119


Findtheequationoftheplaneperpendiculartotheline

andpassingthroughtheorigin.


( − 1, 3, 2)

x + 2y + 3z = 5and3x + 3y+ z = 0.

=

= and

= = .

x − 54

y − 74

z + 3−5

x − 87

y − 41

z− 53

(1, 0, − 1)and(3, 2, 2)

x − 1 = = .1 − y2

z − 23

= =x − 12

y + 2−3

z

5x − y + z + 2 = 0.

= =x− 12

y − 3−1

z − 42

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120


Findtheequationoftheplanepassingthroughtheline

andpoint


121


Findtheanglebetweentheline

andthenormaltotheplane


122


Findtheequationoftheplanewhichpassesthroughthepoint andwhichisattheminimumdistancefromthepoint



= =x − 15

y + 26

z − 34

(4, 3, 7).

→r = (→i + 2→j − →k )+ λ(→i − →j +→k )

→r

.

(2→i −→j + →k ) = 4.

(1, 2, 3)( − 1, 0, 2).

− 1 + 1 − 2

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123 Findthedirectionratiosoforthogonalprojectionof line

intheplane alsofindthedirectionratiosof the imageof thelineintheplane.


124


Find the equation of a plane which is parallel to the plane andwhosedistancefromthepoint is1.


125


Find theequationofaplanewhichpasses through thepoint andwhich isequallyinclinedtotheplanes


126


Find the equation of the image of the plane in plane


127


Findtheequationoftheplanethroughthepoints andparalleltothex-axis.

= =x − 11

y + 1−2

z − 23

x − y + 2z − 3 = 0.

x − 2y + 2z = 5(1, 2, 3)

(1, 2, 3)

x − 2y + 2z − 3 = 0and8x− 4y + z − 7 = 0.

x − 2y + 2z − 3 = 0x + y + z − 1 = 0.

(23, 1)and(4, − 5, 3)

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128


Find the vector equation of the line passing through and parallel to theplanes


129


Findthevalueof forwhichthestraightline

isparalleltotheplane


130


Showthatthelines

and

arecoplanar.


131


Ifthedirectioncosinesofavariablelineintwoadjacentpointsbe

thesmallangle asbetweenthetwopositionsisgivenby


(1, 2, 3)

→r

.

î − ĵ + 2k̂and→r

.

3 î + ĵ + k̂= 6.

m3x− 2y + z + 3 = 0 = 4x+ 3y + 4z + 1

2x − y + mz − 2 = 0.

=

=

x− a + dα − δ

y − aα

z − a − dα + δ

=

=

x − b + cβ − γ

y − bβ

z− b − cβ + γ

l,M , n and l + δl,m + δm+ n + δn

δθ

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132


Findtheequationof theplanethroughthepoints andparallel

totheline .


133


A variable plane passes through a fixed point and meets the axes at show that the locusof thepointof intersectionof theplanes through

paralleltothecoordinateplanesis


134


Showthatthestraightlineswhosedirectioncosinesaregivenbytheequations

areparallelorperpendicularas


135


The perpendicular distance of a corner of uni cube from a diagonal not passing

(1, 0, − 1), (3, 2, 2)

= =x − 11

y − 1−2

z − 23

(α, β, γ)A, B, andC.A, BandC αx−1 + βy −1 + γz −1 = 1.

al + bm + cn = 0and –̂–– 2

+ zm2 = vn2 + wn2 = 0

+ + = 0 or a2(v

+ w) + b2(w + u)

+ c2(u + v) = 0.

a2

u

b2

v

c2

w

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throughitis


136


A point moves on a plane A plane through and

perpendicularto meetsthecoordinateaxesat Iftheplanesthroughparalleltotheplanes respectively,intersectat

findthelocusof


137


Iftheplanes

passthroughastraightline,thenfindthevalueof


138


is a point and are the perpendicular formplanes. If makes angles with the plane and the

planes,respectively,thenprovethat


P + + = 1.x

a

y

b

z

cP

OP A, BandC .A, BandC x = 0, y = 0andz = 0,Q, Q.

x − cy − bz = 0, cx = y+ az = 0andbx + ay − z= 0

a2 + b2 + c2 + 2ab ⋅

P PMandPN P → z − xandx− yOP θ, α, βandγ OMN

x − y, y − zandz − xcos ec2θ = cos ec2α+ cos ec2β + cos ec2γ .

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139


Avariableplane

aredirectioncosinesofnormal intersectsthecoordinateaxesatpoints ,respectively. Show that the foot of the normal on the plane from the origin is theorthocenter of triangle and hence find the coordinate of the circumcentre oftriangle


140


Let

betheequationsoftheplanessuchthat

Thenshowthatthereisacommonlineofintersectionofthethreegivenplanes.


141


Letaplane

are parameters, make an angle with the line with the lineand with theplane Thedistanceof theplane frompoint

is3units.Findthevalueof andtheequationoftheplane.


142


Provethatforallvaluesof ,theplanes

and

lx +my + nz = p(wherel,m, n

) A,BandC

ABCABC.

x − y sin α − zs ∈ β = 0, xs∈ α = zs ∈ γ − y= 0andx sin β+ y sinγ − z= 0

α + β + γ = π/2(whereα,

βandγ ≠ 0).

ax + by + cz + 1 = 0,wherea, b, c

600 x = y = z, 450

x = y − z = 0 θ x = 0.(2, 1, 1) θ

λandμ

+ + − 1

+ λ( − − − 2)= 0

2xa

y

b

2zc

x

a

2yb

z

c

4x 3y

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intersectonthesameline.


143


If isanypointon theplane isapointon the line

suchthat ,thenfindthelocusofthepoint


144


Ifavariableplaneformsatetrahedronofconstantvolume withtheco-ordinateplanes,thenthelocusofthecentroidofthetetrahedronis:


145


Provethatthevolumeoftetrahedronboundedbytheplanes


+ − 5

+ μ( − + 3) = 0

4xa

3yb

5yb

4zc

P lx + my + nz = pandQ OPOP

.OQ = p2 Q.

64k3

→r

.

mĵ + nk̂ = 0,→r.

nk̂ + l î

= 0,→r.

l î + mĵ = 0,

→r

.

l î + mĵ + nk̂ = πs2p3

3lmn

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146


In a three-dimensional space , the equation represents a.Pointsb.planesc.curvesd.pairofstraightlines


147


The line intersects the curve if is

equaltoa. b. c. d.noneofthese


148


Lettheequationsofalineandplanebe

respectively,thena.thelineisparalleltotheplaneb.thelineisperpendiculartotheplanec.thelineliesintheplaned.noneofthese


149


Thelengthoftheperpendicularformtheorigintotheplanepassingthroughthepoint

and containing the line is a. b.

c. d.


xyz x2 − 5x + 6 = 0

= =x − 23

y + 12

z− 1−1

xy = c2, z = 0 c

±1 ±1/3 ±√5

=

= and4x − 2y − z

= 1,

x + 32

y − 43

z + 52

a→r =

→b + λ→c

[→a→b →c ]∣∣∣→a ×

→b +

→b × →c + →c

×→a ∣∣∣

[→a→b →c ]∣∣∣→a ×

→b +

→b ×→c ∣∣∣

[→a→b →c ]∣∣∣→b ×→c + →c × →a ∣∣∣

[→a→b →c ]∣∣∣→c × →a +→a ×

→b∣∣∣

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150


Thedistanceofpoint fromtheline through whichmakesequalangleswiththeaxesisa. b. c. d.


151


TheCartesianequationoftheplane

isa. b. c. d.


152


Aunitvectorparalleltotheintersectionoftheplanes

a. b. c. d.



Let betheline

A( − 2, 3, 1) PQ P( − 3, 5, 2),2/√3 14/√3 16/√3 5/√3

→r = (1 + λ − μ) î

+ (2 − λ) ĵ

+ (3 − 2λ + 2μ)k̂2x + y = 5 2x − y = 5 2x + z = 5 2x − z = 5

→r

.

î − ĵ + k̂

= 5and→r.

2 î + ĵ − 3k̂ = 42 î + 5ĵ − 3k̂

√38

−2 î + 5ĵ − 3k̂

√38

2î + 5ĵ − 3k̂

√38

−2î − 5ĵ − 3k̂

√38

L1

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153

andlet betheline

.Let betheplanewhichcontainstheline andisparallelto Thedistanceof

theplane fromtheoriginisa. b. c. d.noneofthese


154


Fortheline whichoneofthefollowingisincorrect?a.

it liesintheplane b. it issameas line c. itpasses

through d.itisparallelttheplane


155


Thevalueof forwhichstraightlein

isparalleltotheplane isa. b. c. d.



→r 1 = 2î + ĵ − k̂

+ λ(i + 2k̂)L2

→r 2 = 3î + ĵ

+ μ(i + ĵ − k̂)π L1 L2 .

π √6 1/7 √2/7

= = ,x − 11

y − 22

z − 33

x − 2y + z = 0 = =x

1y

2z

3(2, 3, 5) x − 2y + z − 6 = 0

m3x− 2y + z + 3 = 0 = 4x− 3y + 4z + 1

2x − y + mz − 2 = 0 −2 8 −18 11

. .

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156

Theinterceptmadebytheplane onthex-axisisa. b. c.

d.


157


Equationofa line in theplane which isperpendicular to

the line whseequation is andwhich passes through

the point of intersection of is a. b.

c. d.


158


Ifthefootoftheperpendicularfromtheorigintoplaneis ,theequationof

the plane is a. b. c.

d.


159


Theequationoftheplanewhichpassesthroughthepointofintersectionoflines

and at greatest distance from point is a. b.c. d.



Let bepointsontwoskewlines

→r

.→n = q

q

î

.→n

î

.→n

q

î

.→n

q

q

∣∣→n ∣∣

π ≡ 2x− y + z − 4 = 0

l = =x − 21

y − 2−1

z − 3−2

landπ = =x − 21

y − 15

z − 1−1

= =x − 13

y − 35

z − 5−1

= =x+ 22

y + 1−1

z + 11

= =x − 22

y − 1−1

z − 11

P(a, b, c)

= = = 3x

a

y

b

z

cax + by + cz = 3

ax + by + cz = a2 + b2 + c2 ax + by + cz = a + b + c

= = ,

and =

=

x − 13

y − 21

z − 32

x − 31

y − 12

z − 23

(0, 0, 0) 4x + 3y + 5z = 254x+ 3y = 5z = 50 3x+ 4y + 5z = 49 x + 7y − 5z = 2

A(→a )andB(→b )

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160

and the shortest distance between the skew lines is are unitvectors forming adjacent sides of a parallelogramenclosing an area of 1/2 units. If

anglebetween andthelineofshortestdistanceis then a. b. c.

d.


161


Let

be three points, then equation of a plane parallel to the plane which is atdistance is a. b. c.

d.


162


Thepointontheline atadistanceof6 fromthepoint

isa. b. c. d.noneofthese


163


Thecoordinatesothefootoftheperpendiculardrawnfromtheorigintothelinejoiningthepoint willbea. b. c. d.noneofthese

→r = →a + λ→p and→r =

→b

+ u→q1, where→p and→q

AB 600, AB =12

2

1 λR = {10}

A(1, 1, 1), B(2, 3, 5)andC(

− 1, 0, 2)ABC

2 2x − 3y + z + 2√14 = 0 2x − 3y + z − √14 = 02x− 3y + z + 2 = 0 2x − 3y + z− 2 = 0

= =x − 21

y + 3−2

z+ 5−2

(2, − 3, − 5) (3, − 5, − 3) (4, − 7, − 9) 0, 2, − 1

( − 9, 4, 5)and(10, 0, − 1) ( − 3, 2, 1) (1, 2, 2) 4, 5, 3

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164


If and arethree non-coplanar vectors, then three lines , ; , ;

are


165


The length of projection of the line segment joining the points on the plane is equal to a. b.

c. d.


166


Thenumberofplanesthatareequidistantfromfournon-coplanarpointsisa. b. c.d.


167


In a three-dimensional coordinate system, are images of a pointinthe planes,respectively.If isthecentroidof

triangle thenareaoftriangle is( istheorigin)a. b. c.

d.noneofthese


P1 :→r . →n 1 − d1 = 0 P2 :

→r . →n 2 − d2 = 0 P3 :

→r . →n 3 − d3 = 0

P1 = 0 P2 = 0 P2 = 0 P3 = 0P3 = 0 P1 = 0

(1, 0, − 1)and( − 1, 2, 2) x + 3y − 5z = 6 2

√ 27153

√ 47231

√ 47435

3 47 9

P , Q, andRA(a, b, c) x − y, y − zandz − x G

PQR, AOG O 0 a2 + b2 + c2

(a2 + b2 + c2)23

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168


A plane passing through cuts positive direction of coordinates axes at

thenthevolumeoftetrahedron satisfiesa. b.

c. d.noneofthese


169


Iflines andthird linepassingthrough forma

triangleofarea units, then thepointof intersectionof third linewith thesecond

linewillbea. b. c. d.noneofthese


170


Thepointofintersectionofthelinepassingthrough andintersectingthelines

and with plane is a. b. c.

d.



(1, 1, 1)

A, BandC, OABC V ≤92

V ≥92

V =92

x = y = zandx = =y

2z

3(1, 1, 1)

√6

(1, 2, 3) 2, 4, 6 , ,4363123

(0, 0, 1)x + 2y + z = 1, − x + y− 2z = 2

x + y = 2, x + z = 2 xy ( , − , 0)53

13

(1, 1, 0)

( , , 0)2313

( − , , 0)5313

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171

Shortestdistancebetweenthelines

isequaltoa. b. c. d.noneofthese


172


Distance of point from the plane is a. b. c.

d.noneofthese


173


Thereflectionof thepoint in theplane isa. b.

c. d.noneofthese


=

= and

= =

x − 11

y − 11

z − 11

x − 21

y − 31

z− 41

√14 √7 √2

P(→p ) →r.→n = 0

∣∣∣→p

.→n∣∣∣

∣∣→p × →n ∣∣∣∣→n ∣∣

∣∣∣→p

.→n∣∣∣

∣∣→n ∣∣

→a

→r

.→n = q →a +

(→q −→a.→n)

∣∣→n ∣∣

→a + 2

⎛⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎠→n

(→q − →a.→n)

∣∣→n ∣∣

→a + →n

2(→q +→a.→n)

∣∣→n ∣∣2

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174


Line willnotmeettheplane ifa.

b. c. d.


175


Ifalinemakesanangleof withthepositivedirectionofeachofx-axisandy-axis,

thentheangelthatthelinemakeswiththepositivedirectionofthez-axisisa. b.

c. d.


176


The ratio in which the plane divides the line joining thepoints

isa. b. c. d.


177


The image of the point in the plane is a.

b. c. d.

→r = →a + λ

→b

→r

.→n = q,

→b

.→n = 0, →a

.→n = q

→b

.→n ≠ 0, →a

.→n ≠ q

→b

.→n = 0,→a

.→n ≠ q

→b

.→n ≠ 0, →a

.→n = q

π

4π

3π

4π

2π

6

→r

.→i − 2

→j + 3

→k = 17

−2→i + 4

→j + 7

→k and3

→i

− 5→j + 8

→k

1: 5 1 : 10 3: 5 3 : 10

( − 1, 3, 4) x − 2y = 0

( − , − , 4)173

193

(15, 11, 4) ( − , − , 1)173

193

( , , 4)95−135

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178


Thedistancebetweentheline

andtheplane isa. b. c. d.


179


Let bethelineofintersectionoftheplanes

If makesanangle withthepositivex-axis,then equalsa. b. c. d.


180


The length of the perpendicular drawn from to the line

isa. b. c. d.


→r = 2 î − 2ĵ + 3k̂

+ λ(î − ĵ + 4k̂)→r

.

î + 5ĵ + k̂ = 510

3√3

109

103

310

L2x+ 3y + z = 1andx + 3y+ 2z = 2.L α cosα

12

11

√21

√3

(1, 2, 3)

= =x − 63

y − 72

z − 7−2

4 5 6 7

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181


If angle bertween the line and the plane

issuchthat thevalueof isa. b.

c. d.


182


Theintersectionofthespheres

isthesameastheintersectionofoneofthespheresandtheplanea.b. c. d.


183


Aplanemakesintercepts whosemeasurementsare and on

the axes. The area of triangle is a. b.

c. d.


184


Alinemakesanangel witheachofthex-andz-axes.Iftheangel whichitmakes

withthey-axis,issuchthat ,then equalsa. b. c. d.


185


Theshortestdistancefromtheplane tothesphere

isa. b. c. d.

θ = =x + 11

y − 12

z − 22

2x− y +√λz + 4 = 0 s∫hη = 1/3, λ − 35

53

−43

34

x2 + y2 + z2 + 7x − 2y − z

= 13andx2 + y2 + z2 − 3x+ 3y + 4z = 8

x − y − z = 1x − 2y − z = 1 x − y − 2z = 1 2x− y − z = 1

OA,OBandOC a, b c

OX, OY andOZ ABC (ab + bc + ca)12

abc(a + b + c)12

(a2b2 + b2c2 + c2a2)1/ 21

2(a + b + c)212

θ β,

sin2 β = 3 sin2 θ cos2 θ23

15

35

25

12x + 4y + 3z = 327x2 + y2 + z2 + 4x − 2y− 6z = 155

39 26 41 −413

13

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186


Atetrahedronhasvertices

thenanglebetween face willbea. b. c. d.


187


Theradiusofthecircleinwhichthesphere

iscutbytheplane isa. b. c. d.


O(0, 0, 0),A(1, 2, 1),

B(2, 1, 3), andC( − 1, 1, 2),

OABandABC cos−1( )1731

300 900

cos−1( )1935

x2 + y2 + z2 + 2z − 2y − 4z− 19 = 0

x + 2y + 2z + 7 = 0 2 3 4 1

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188


Thelines

are coplanar if a. b. c. d.


189


Thepointofintersectionofthelines

isa. b. c. d.


190


Two systems of rectangular axes have the same origin. If a plane cuts them atdistance and fromtheorigin,thena.

b.

c.

d.

=

= a

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CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER ......Find the vector equation of line passing through...

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