--

-----

~~t o~ ~I

f-~~ (lit if) middot KS+e) r ( c+A)J ~ 2 fl I C )

bi f ID (B 1- tj (t lr i Af- p pound r 1) ([ t1 F) c-middot I[A l))

lIi tR lif ACt ~ It II -r e 1 ~ )

-- Amiddot lR C) + A~ + ~ =---yen

+ $~+~ + 15 (CIIA)

- A WAC) + 15- (Cft fJ)

- A-(3lCj + t LBfc) - ~() (l3fc)

I A f1

V-Y~s -~ Vt lt--~

~((Ef-~) f -tLxtK~)-+ ~x 0--1ltJ= V

A Y- aXC) B C19- c) - c 19-ii)

BX (g 11 A) - C Cs5- Ii) - fll73- c)

C X (AIV - Ii le-)- 7 (c )

~~~Zl~~

A I

_~ L~-tJJ-B -- )(l -r ~ Jl _ l- l shy

fJ Q -- I~+ Yl 1

BJ ~ - ~ r v + f)) j Ie[~ 0(

-x~ B c)feI~amp)) IL l(4te5o(

B ~ - lAI1S ~ 5[o+P)t f 0( + (Ilmiddot)f -- oIr(oNamp) 9

A 5 c rIi f ~ ~ )~ -r- isgt - c(~ - [4 ~ I ( r

---

8y

n~ - -K f r i ) I ( 1Ld + J ) i ~ yen - ~ -t l k iJ( ~to) It ~ )111) ~ If- fDl-+ f)) (f R - ptgtlt (Jd+ 9)] ~

~ r lt If rr ~ I p9 ~

~ ~ t_gt- ~ ( fc) ~lr-~ J~fV ~

2 l) -z K - 2 [f(~)J - f(p) lampl + (fcf))

- [rri811+ [frt9)]l 1J~~

-f()l)

fk

cJl ~ ~t9JV +(c~5f)dt9 )) ~ c)l)J(- Xn--f)df7

K

J

C-flt c f ~) J - -shy

~ I() f T ttr)~de A~ - _ e

-) f

f (~) f - fCb) ~ - ~ - 0- Ji~ - -J9~ )

---o J ~ ~

The DOT and CROSS PRODUCT 32

63 Find the projection of the vector 21 - 3j + 6k OD the vector 1 + 2j + 21t Ails 83

64 Find the projection of the vector 41 - 3j + k on the Une passing through the pOints (23-1) and (-2-43) AilS 1

85 If A 41 - j + 31t and 8 -21 + j - 21t find a unit vector perpendicular to both A and 8 AilS plusmn (I - 2j - 2k)3

61 Find the acute angle formed by two diagonals of a cUbe

67 Find a unit vector parallel to the xy plane and perpendicular to the vector 41- 3j +k AilS plusmn (31 +4j)5

68 Show that A =(21- 2j +1t)3 8 (I + 2j + 21t)3 and C = (21+ j - 2k)3 are mutually orthogonal unit vectors

69 Find the worlt done in moving an object along a straight line from (32-1) to (2-14) in a force field given by F = 41-3j+2k AilS 15

70 Let F be a constant vector force field Show that the work done in moving an object around any closed polshyygon in this force field is Zero

71 Prove that an angie inscribed in a semi-circle is a right angle

72 Let ABCD be a parallelogram Prove that AB2 + BC2 + CD2 + DA2 AC2 + BD2

73 If ABCD is any quadrilateral and P and Q are the midpoints of its diagonals prOVe that

AB2 + BC2 + CD2 + DA2 = AC2 + BD2 + 4PQ2 This is a generalization of the preceding problem

74 (a) Find an equation of a plane perpendicular to a given vector A and distant p from the origin (b) Express the eQuation of (a) in rectangular coordinates AilS (a) rmiddotn = p where D AA (b) A 1 x + ~y + Aaz = Ap

75 Let rl and r2 be unit vectors in the xy plane making angles a and 13 with the positive x-axis (a) Prove that r1 cos a 1 + sin a 1 r2 = cos f3 I + sin f3 j (b) By considering rl r2 prove the trigonometric formulas

cos (a - f3) = cos a cos f3 + sin a sin 13 cos (a +3) = cos a cosS - sin a sin S

71 Let a be the position vector of a given point (xlYl 1) and r the position vector of any point (xy ) Deshyscribe the locus ofrlf (a) Ir-al = 3 (b) (r-a)middota=O (e) (r-o)middotr=O AilS (a) Sphere centre at (Xl 11 1) and radius 3

(b) Plane perpendicular to a and passing through its terminal pOint (e) Sphere with centre at (x2 Y12 z2) and radius I x~ + Y~ + z~ or a sphere with a as diameter

77 Given that A = 31+j+21t and 8 1-2j-41t are the position vectors of points P and Q respectively (a) Find an eQuation for the plane passing through Q and perpendicular to line PQ (b) What is the distance from the point (-111) to the plane AilS (a) (r-8)(A-8) 0 or 2x+3y+6z -28 (b) 5

78 Evaluate each of the following (a) 2jx(31-4k) (b) (1+2j)xlt (e) (21-41t)x(I+2j) (d) (41+j-2k)x(31+1t) (e) (21+j-It)x(31-2j+4k) AilS (a)-81-6It (b) 21-j (e) 81-41+4k (d) I-I0j-3k (e) 21-11j-7k

79 If A =31-j-2k and B 21+3j+k find (a) IAxBI (b) (A+28)x(2A-8) (e) (A+8)x(A-B)I AilS (a) Yi95 (b) -251 +35j-5511 (e) 2ms

80 If A 1-2j-3k 8 21+j-1I and C = i+3j-2k find (e) I (A x8) x C I (e) A (8 xC) (e) (A xB) x (B xC) (b) IA x (B xC) I (d) (A xB)middot C (f) (A xB)(8middot C) AilS (a)5V2s (b31iO (e)-20 (d)-20 (e)-401-20j+201l laquof)351-35j+35k

81 Show that If A~ 0 and both of the conditions (a) A-B = AmiddotC and (b) Ax8 AxC hold simultaneously then B C but if 0nIy one of these conditions holds then 8 ~ C necessarily

82 Find the area of a parallelogram having diagonals A 31 +j - 2k and B = i - 3j + 4k AilS 5 v3

33 The DOT and CROSS PRODUCT

83 F1nd the area of a triangle with vertices at (3-12) 0-1-3) and (4-31) Ans i Ii65

84 It A = 21 + J - 3k and B 1- 2j +k find a vector of magnUude 5 perpendicular to both A and B

Ans plusmn 5f(I+J+k)

85 Use Problem 75 to derive the formulas s1n(ltl-t3) sinltlcost3 - cosltlslnt3 sin(ltl+t3) sinltlcost3 + cosltls1n~

86 A force given by F 31 + 2j - 4k is applied at the point (1 -12) Find the moment of F about the point (2 -I 3) Ans 21 - 1J - 2k

87 Tbe angular velocity of a rotating rigid body about an axis of rotation is given by Col 41 + J - 2k Find the linear velocity of a po1nt P on the body whose position vector relative to a point on the axis of rotation is 21-3J+k AlIs -51- 8J - 14k

88 Simplify (A+B)middot(B+C)x(C+A) Ans 2ABxC

Aa Ab Ac

89 Provethat (A BxC)(amiddotbxc) Bmiddota Bb Bmiddotc

Cmiddota Cmiddotb Cmiddotc

98 Find tbe volume of the parallelepiped whose edges are represented by A 21 - 3j +4k B =1 + 2J - k C 31- J + 2k Ans 7

91 It A B xC 0 show that either (0) A Band C are coplanar but no two of them are colllnear or (b) two of tbe vectors A B and C are colllnear or (c) all of tbe vectors A Band C are colllnear

92 Find tbe constant 0 such that the vectors 21-J + k 1 + 2J - 3t and 31 +aj + 5k are coplanar Ans 0 = -4

13 It A =a + 11b + z1C B 2a + 12b + Z2C and C sa + 1s b + sc prove that

1 11 1 ABxC 2 12 2 (amiddotbxc)

s 1s s

94 Prove that a necessary and sufficient condition tbat A x (B xC) (AxB)xC is (Ax C) x B = O Disshycuss the cases where A B 0 or Bmiddot C O

95 Let pOints P Q and R have position vectors r1 31 - 2j - k r2 i+ 3j +4k and r s = 21 + j - 2k relative to an origin O Find tbe distance from P to the plane OQR Ans 3

96 Find the shortest distance from (6-44) to the line joining (212) and (3-14) Ans 3

97 Given points P(213) Q(121) R(-1-2-2) and SO-40) find the shortest distance between lines PQ and RS Ans 3v2

98 Prove that the perpendiculars from the vertices of a triangle to the opposite sides (extended if necessary) meet in a point (theorthocentre of the triangle)

99 Prove that tbe perpendicular bisectors of the sides of a triangle meet in a point (the circumcentre of the trishyangle)

100 Prove that (AxB)middot(CxD) + (BxC)middot(AxD) + (CxA)middot(BxD) = o

101 Let PQR be a spherical triangle whose sides pqT are arcs of great circles Prove the law of cosirampe8 for sphericol triangles

cos p = cos q cos T + sin q sin r cos P

with analogous formulas for cos q and cos r obtained by cyclic permutation of tbe letters [Hlnt Interpret both aides of the identity (A xB) (A xC) (B C)(A A) - (A C)(B A)]

3If middot

COOOCR JEDOU DKCR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull Monday Ma bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullllABDALLMOHAMEDSEBA~G 71213181 13MAY bullbullbullbullbullbullbullbullbullbullbullbullbull~ bullbull 1~3 2013 112 AM 21MOHAMEDSOADKHALEDABHMAHMOUDABDALLAMR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull

~ 337 C 05JUN RUHC~~L~TIFMOHAMEDMRS E WEFILED3~~~ Di~UG CAIRUH H~~ ii~~ 0030 0 E SA VENDOR L EXISTS 1440 0 VENDOR R~CATOR DATA EXISTS gtFF SERVICE I~~DATA EXISTS gtVL CUSTOM CHECK TION EXISTS gtVR

NE-RUHTALT RULES EXIST gtSI

G~~IW966~~i~v~FG~~P REF CRI~RU OTAL 1033000

31- ( r -I

S =- ~--b J

J shy

~poundv fY)N 0 c ~(~-gt) l2

1

0

~ - t c -- ----shy

)t ) -- ----~

f-Y~[8

A I

_~ L~-tJJ-B -- )(l -r ~ Jl _ l- l shy

fJ Q -- I~+ Yl 1

BJ ~ - ~ r v + f)) j Ie[~ 0(

-x~ B c)feI~amp)) IL l(4te5o(

B ~ - lAI1S ~ 5[o+P)t f 0( + (Ilmiddot)f -- oIr(oNamp) 9

A 5 c rIi f ~ ~ )~ -r- isgt - c(~ - [4 ~ I ( r

---

8y

n~ - -K f r i ) I ( 1Ld + J ) i ~ yen - ~ -t l k iJ( ~to) It ~ )111) ~ If- fDl-+ f)) (f R - ptgtlt (Jd+ 9)] ~

~ r lt If rr ~ I p9 ~

~ ~ t_gt- ~ ( fc) ~lr-~ J~fV ~

2 l) -z K - 2 [f(~)J - f(p) lampl + (fcf))

- [rri811+ [frt9)]l 1J~~

-f()l)

fk

cJl ~ ~t9JV +(c~5f)dt9 )) ~ c)l)J(- Xn--f)df7

K

J

C-flt c f ~) J - -shy

~ I() f T ttr)~de A~ - _ e

-) f

f (~) f - fCb) ~ - ~ - 0- Ji~ - -J9~ )

---o J ~ ~

The DOT and CROSS PRODUCT 32

63 Find the projection of the vector 21 - 3j + 6k OD the vector 1 + 2j + 21t Ails 83

64 Find the projection of the vector 41 - 3j + k on the Une passing through the pOints (23-1) and (-2-43) AilS 1

85 If A 41 - j + 31t and 8 -21 + j - 21t find a unit vector perpendicular to both A and 8 AilS plusmn (I - 2j - 2k)3

61 Find the acute angle formed by two diagonals of a cUbe

67 Find a unit vector parallel to the xy plane and perpendicular to the vector 41- 3j +k AilS plusmn (31 +4j)5

68 Show that A =(21- 2j +1t)3 8 (I + 2j + 21t)3 and C = (21+ j - 2k)3 are mutually orthogonal unit vectors

69 Find the worlt done in moving an object along a straight line from (32-1) to (2-14) in a force field given by F = 41-3j+2k AilS 15

70 Let F be a constant vector force field Show that the work done in moving an object around any closed polshyygon in this force field is Zero

71 Prove that an angie inscribed in a semi-circle is a right angle

72 Let ABCD be a parallelogram Prove that AB2 + BC2 + CD2 + DA2 AC2 + BD2

73 If ABCD is any quadrilateral and P and Q are the midpoints of its diagonals prOVe that

AB2 + BC2 + CD2 + DA2 = AC2 + BD2 + 4PQ2 This is a generalization of the preceding problem

74 (a) Find an equation of a plane perpendicular to a given vector A and distant p from the origin (b) Express the eQuation of (a) in rectangular coordinates AilS (a) rmiddotn = p where D AA (b) A 1 x + ~y + Aaz = Ap

75 Let rl and r2 be unit vectors in the xy plane making angles a and 13 with the positive x-axis (a) Prove that r1 cos a 1 + sin a 1 r2 = cos f3 I + sin f3 j (b) By considering rl r2 prove the trigonometric formulas

cos (a - f3) = cos a cos f3 + sin a sin 13 cos (a +3) = cos a cosS - sin a sin S

71 Let a be the position vector of a given point (xlYl 1) and r the position vector of any point (xy ) Deshyscribe the locus ofrlf (a) Ir-al = 3 (b) (r-a)middota=O (e) (r-o)middotr=O AilS (a) Sphere centre at (Xl 11 1) and radius 3

(b) Plane perpendicular to a and passing through its terminal pOint (e) Sphere with centre at (x2 Y12 z2) and radius I x~ + Y~ + z~ or a sphere with a as diameter

77 Given that A = 31+j+21t and 8 1-2j-41t are the position vectors of points P and Q respectively (a) Find an eQuation for the plane passing through Q and perpendicular to line PQ (b) What is the distance from the point (-111) to the plane AilS (a) (r-8)(A-8) 0 or 2x+3y+6z -28 (b) 5

78 Evaluate each of the following (a) 2jx(31-4k) (b) (1+2j)xlt (e) (21-41t)x(I+2j) (d) (41+j-2k)x(31+1t) (e) (21+j-It)x(31-2j+4k) AilS (a)-81-6It (b) 21-j (e) 81-41+4k (d) I-I0j-3k (e) 21-11j-7k

79 If A =31-j-2k and B 21+3j+k find (a) IAxBI (b) (A+28)x(2A-8) (e) (A+8)x(A-B)I AilS (a) Yi95 (b) -251 +35j-5511 (e) 2ms

80 If A 1-2j-3k 8 21+j-1I and C = i+3j-2k find (e) I (A x8) x C I (e) A (8 xC) (e) (A xB) x (B xC) (b) IA x (B xC) I (d) (A xB)middot C (f) (A xB)(8middot C) AilS (a)5V2s (b31iO (e)-20 (d)-20 (e)-401-20j+201l laquof)351-35j+35k

81 Show that If A~ 0 and both of the conditions (a) A-B = AmiddotC and (b) Ax8 AxC hold simultaneously then B C but if 0nIy one of these conditions holds then 8 ~ C necessarily

82 Find the area of a parallelogram having diagonals A 31 +j - 2k and B = i - 3j + 4k AilS 5 v3

33 The DOT and CROSS PRODUCT

83 F1nd the area of a triangle with vertices at (3-12) 0-1-3) and (4-31) Ans i Ii65

84 It A = 21 + J - 3k and B 1- 2j +k find a vector of magnUude 5 perpendicular to both A and B

Ans plusmn 5f(I+J+k)

85 Use Problem 75 to derive the formulas s1n(ltl-t3) sinltlcost3 - cosltlslnt3 sin(ltl+t3) sinltlcost3 + cosltls1n~

86 A force given by F 31 + 2j - 4k is applied at the point (1 -12) Find the moment of F about the point (2 -I 3) Ans 21 - 1J - 2k

87 Tbe angular velocity of a rotating rigid body about an axis of rotation is given by Col 41 + J - 2k Find the linear velocity of a po1nt P on the body whose position vector relative to a point on the axis of rotation is 21-3J+k AlIs -51- 8J - 14k

88 Simplify (A+B)middot(B+C)x(C+A) Ans 2ABxC

Aa Ab Ac

89 Provethat (A BxC)(amiddotbxc) Bmiddota Bb Bmiddotc

Cmiddota Cmiddotb Cmiddotc

98 Find tbe volume of the parallelepiped whose edges are represented by A 21 - 3j +4k B =1 + 2J - k C 31- J + 2k Ans 7

91 It A B xC 0 show that either (0) A Band C are coplanar but no two of them are colllnear or (b) two of tbe vectors A B and C are colllnear or (c) all of tbe vectors A Band C are colllnear

92 Find tbe constant 0 such that the vectors 21-J + k 1 + 2J - 3t and 31 +aj + 5k are coplanar Ans 0 = -4

13 It A =a + 11b + z1C B 2a + 12b + Z2C and C sa + 1s b + sc prove that

1 11 1 ABxC 2 12 2 (amiddotbxc)

s 1s s

94 Prove that a necessary and sufficient condition tbat A x (B xC) (AxB)xC is (Ax C) x B = O Disshycuss the cases where A B 0 or Bmiddot C O

95 Let pOints P Q and R have position vectors r1 31 - 2j - k r2 i+ 3j +4k and r s = 21 + j - 2k relative to an origin O Find tbe distance from P to the plane OQR Ans 3

96 Find the shortest distance from (6-44) to the line joining (212) and (3-14) Ans 3

97 Given points P(213) Q(121) R(-1-2-2) and SO-40) find the shortest distance between lines PQ and RS Ans 3v2

98 Prove that the perpendiculars from the vertices of a triangle to the opposite sides (extended if necessary) meet in a point (theorthocentre of the triangle)

99 Prove that tbe perpendicular bisectors of the sides of a triangle meet in a point (the circumcentre of the trishyangle)

100 Prove that (AxB)middot(CxD) + (BxC)middot(AxD) + (CxA)middot(BxD) = o

101 Let PQR be a spherical triangle whose sides pqT are arcs of great circles Prove the law of cosirampe8 for sphericol triangles

cos p = cos q cos T + sin q sin r cos P

with analogous formulas for cos q and cos r obtained by cyclic permutation of tbe letters [Hlnt Interpret both aides of the identity (A xB) (A xC) (B C)(A A) - (A C)(B A)]

3If middot

COOOCR JEDOU DKCR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull Monday Ma bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullllABDALLMOHAMEDSEBA~G 71213181 13MAY bullbullbullbullbullbullbullbullbullbullbullbullbull~ bullbull 1~3 2013 112 AM 21MOHAMEDSOADKHALEDABHMAHMOUDABDALLAMR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull

~ 337 C 05JUN RUHC~~L~TIFMOHAMEDMRS E WEFILED3~~~ Di~UG CAIRUH H~~ ii~~ 0030 0 E SA VENDOR L EXISTS 1440 0 VENDOR R~CATOR DATA EXISTS gtFF SERVICE I~~DATA EXISTS gtVL CUSTOM CHECK TION EXISTS gtVR

NE-RUHTALT RULES EXIST gtSI

G~~IW966~~i~v~FG~~P REF CRI~RU OTAL 1033000

31- ( r -I

S =- ~--b J

J shy

~poundv fY)N 0 c ~(~-gt) l2

1

0

~ - t c -- ----shy

)t ) -- ----~

f-Y~[8

~ ~ t_gt- ~ ( fc) ~lr-~ J~fV ~

2 l) -z K - 2 [f(~)J - f(p) lampl + (fcf))

- [rri811+ [frt9)]l 1J~~

-f()l)

fk

cJl ~ ~t9JV +(c~5f)dt9 )) ~ c)l)J(- Xn--f)df7

K

J

C-flt c f ~) J - -shy

~ I() f T ttr)~de A~ - _ e

-) f

f (~) f - fCb) ~ - ~ - 0- Ji~ - -J9~ )

---o J ~ ~

The DOT and CROSS PRODUCT 32

63 Find the projection of the vector 21 - 3j + 6k OD the vector 1 + 2j + 21t Ails 83

64 Find the projection of the vector 41 - 3j + k on the Une passing through the pOints (23-1) and (-2-43) AilS 1

85 If A 41 - j + 31t and 8 -21 + j - 21t find a unit vector perpendicular to both A and 8 AilS plusmn (I - 2j - 2k)3

61 Find the acute angle formed by two diagonals of a cUbe

67 Find a unit vector parallel to the xy plane and perpendicular to the vector 41- 3j +k AilS plusmn (31 +4j)5

68 Show that A =(21- 2j +1t)3 8 (I + 2j + 21t)3 and C = (21+ j - 2k)3 are mutually orthogonal unit vectors

69 Find the worlt done in moving an object along a straight line from (32-1) to (2-14) in a force field given by F = 41-3j+2k AilS 15

70 Let F be a constant vector force field Show that the work done in moving an object around any closed polshyygon in this force field is Zero

71 Prove that an angie inscribed in a semi-circle is a right angle

72 Let ABCD be a parallelogram Prove that AB2 + BC2 + CD2 + DA2 AC2 + BD2

73 If ABCD is any quadrilateral and P and Q are the midpoints of its diagonals prOVe that

AB2 + BC2 + CD2 + DA2 = AC2 + BD2 + 4PQ2 This is a generalization of the preceding problem

74 (a) Find an equation of a plane perpendicular to a given vector A and distant p from the origin (b) Express the eQuation of (a) in rectangular coordinates AilS (a) rmiddotn = p where D AA (b) A 1 x + ~y + Aaz = Ap

75 Let rl and r2 be unit vectors in the xy plane making angles a and 13 with the positive x-axis (a) Prove that r1 cos a 1 + sin a 1 r2 = cos f3 I + sin f3 j (b) By considering rl r2 prove the trigonometric formulas

cos (a - f3) = cos a cos f3 + sin a sin 13 cos (a +3) = cos a cosS - sin a sin S

71 Let a be the position vector of a given point (xlYl 1) and r the position vector of any point (xy ) Deshyscribe the locus ofrlf (a) Ir-al = 3 (b) (r-a)middota=O (e) (r-o)middotr=O AilS (a) Sphere centre at (Xl 11 1) and radius 3

(b) Plane perpendicular to a and passing through its terminal pOint (e) Sphere with centre at (x2 Y12 z2) and radius I x~ + Y~ + z~ or a sphere with a as diameter

77 Given that A = 31+j+21t and 8 1-2j-41t are the position vectors of points P and Q respectively (a) Find an eQuation for the plane passing through Q and perpendicular to line PQ (b) What is the distance from the point (-111) to the plane AilS (a) (r-8)(A-8) 0 or 2x+3y+6z -28 (b) 5

78 Evaluate each of the following (a) 2jx(31-4k) (b) (1+2j)xlt (e) (21-41t)x(I+2j) (d) (41+j-2k)x(31+1t) (e) (21+j-It)x(31-2j+4k) AilS (a)-81-6It (b) 21-j (e) 81-41+4k (d) I-I0j-3k (e) 21-11j-7k

79 If A =31-j-2k and B 21+3j+k find (a) IAxBI (b) (A+28)x(2A-8) (e) (A+8)x(A-B)I AilS (a) Yi95 (b) -251 +35j-5511 (e) 2ms

80 If A 1-2j-3k 8 21+j-1I and C = i+3j-2k find (e) I (A x8) x C I (e) A (8 xC) (e) (A xB) x (B xC) (b) IA x (B xC) I (d) (A xB)middot C (f) (A xB)(8middot C) AilS (a)5V2s (b31iO (e)-20 (d)-20 (e)-401-20j+201l laquof)351-35j+35k

81 Show that If A~ 0 and both of the conditions (a) A-B = AmiddotC and (b) Ax8 AxC hold simultaneously then B C but if 0nIy one of these conditions holds then 8 ~ C necessarily

82 Find the area of a parallelogram having diagonals A 31 +j - 2k and B = i - 3j + 4k AilS 5 v3

33 The DOT and CROSS PRODUCT

83 F1nd the area of a triangle with vertices at (3-12) 0-1-3) and (4-31) Ans i Ii65

84 It A = 21 + J - 3k and B 1- 2j +k find a vector of magnUude 5 perpendicular to both A and B

Ans plusmn 5f(I+J+k)

85 Use Problem 75 to derive the formulas s1n(ltl-t3) sinltlcost3 - cosltlslnt3 sin(ltl+t3) sinltlcost3 + cosltls1n~

86 A force given by F 31 + 2j - 4k is applied at the point (1 -12) Find the moment of F about the point (2 -I 3) Ans 21 - 1J - 2k

87 Tbe angular velocity of a rotating rigid body about an axis of rotation is given by Col 41 + J - 2k Find the linear velocity of a po1nt P on the body whose position vector relative to a point on the axis of rotation is 21-3J+k AlIs -51- 8J - 14k

88 Simplify (A+B)middot(B+C)x(C+A) Ans 2ABxC

Aa Ab Ac

89 Provethat (A BxC)(amiddotbxc) Bmiddota Bb Bmiddotc

Cmiddota Cmiddotb Cmiddotc

98 Find tbe volume of the parallelepiped whose edges are represented by A 21 - 3j +4k B =1 + 2J - k C 31- J + 2k Ans 7

91 It A B xC 0 show that either (0) A Band C are coplanar but no two of them are colllnear or (b) two of tbe vectors A B and C are colllnear or (c) all of tbe vectors A Band C are colllnear

92 Find tbe constant 0 such that the vectors 21-J + k 1 + 2J - 3t and 31 +aj + 5k are coplanar Ans 0 = -4

13 It A =a + 11b + z1C B 2a + 12b + Z2C and C sa + 1s b + sc prove that

1 11 1 ABxC 2 12 2 (amiddotbxc)

s 1s s

94 Prove that a necessary and sufficient condition tbat A x (B xC) (AxB)xC is (Ax C) x B = O Disshycuss the cases where A B 0 or Bmiddot C O

95 Let pOints P Q and R have position vectors r1 31 - 2j - k r2 i+ 3j +4k and r s = 21 + j - 2k relative to an origin O Find tbe distance from P to the plane OQR Ans 3

96 Find the shortest distance from (6-44) to the line joining (212) and (3-14) Ans 3

97 Given points P(213) Q(121) R(-1-2-2) and SO-40) find the shortest distance between lines PQ and RS Ans 3v2

98 Prove that the perpendiculars from the vertices of a triangle to the opposite sides (extended if necessary) meet in a point (theorthocentre of the triangle)

99 Prove that tbe perpendicular bisectors of the sides of a triangle meet in a point (the circumcentre of the trishyangle)

100 Prove that (AxB)middot(CxD) + (BxC)middot(AxD) + (CxA)middot(BxD) = o

101 Let PQR be a spherical triangle whose sides pqT are arcs of great circles Prove the law of cosirampe8 for sphericol triangles

cos p = cos q cos T + sin q sin r cos P

with analogous formulas for cos q and cos r obtained by cyclic permutation of tbe letters [Hlnt Interpret both aides of the identity (A xB) (A xC) (B C)(A A) - (A C)(B A)]

3If middot

COOOCR JEDOU DKCR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull Monday Ma bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullllABDALLMOHAMEDSEBA~G 71213181 13MAY bullbullbullbullbullbullbullbullbullbullbullbullbull~ bullbull 1~3 2013 112 AM 21MOHAMEDSOADKHALEDABHMAHMOUDABDALLAMR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull

~ 337 C 05JUN RUHC~~L~TIFMOHAMEDMRS E WEFILED3~~~ Di~UG CAIRUH H~~ ii~~ 0030 0 E SA VENDOR L EXISTS 1440 0 VENDOR R~CATOR DATA EXISTS gtFF SERVICE I~~DATA EXISTS gtVL CUSTOM CHECK TION EXISTS gtVR

NE-RUHTALT RULES EXIST gtSI

G~~IW966~~i~v~FG~~P REF CRI~RU OTAL 1033000

31- ( r -I

S =- ~--b J

J shy

~poundv fY)N 0 c ~(~-gt) l2

1

0

~ - t c -- ----shy

)t ) -- ----~

f-Y~[8

K

J

C-flt c f ~) J - -shy

~ I() f T ttr)~de A~ - _ e

-) f

f (~) f - fCb) ~ - ~ - 0- Ji~ - -J9~ )

---o J ~ ~

The DOT and CROSS PRODUCT 32

63 Find the projection of the vector 21 - 3j + 6k OD the vector 1 + 2j + 21t Ails 83

64 Find the projection of the vector 41 - 3j + k on the Une passing through the pOints (23-1) and (-2-43) AilS 1

85 If A 41 - j + 31t and 8 -21 + j - 21t find a unit vector perpendicular to both A and 8 AilS plusmn (I - 2j - 2k)3

61 Find the acute angle formed by two diagonals of a cUbe

67 Find a unit vector parallel to the xy plane and perpendicular to the vector 41- 3j +k AilS plusmn (31 +4j)5

68 Show that A =(21- 2j +1t)3 8 (I + 2j + 21t)3 and C = (21+ j - 2k)3 are mutually orthogonal unit vectors

69 Find the worlt done in moving an object along a straight line from (32-1) to (2-14) in a force field given by F = 41-3j+2k AilS 15

70 Let F be a constant vector force field Show that the work done in moving an object around any closed polshyygon in this force field is Zero

71 Prove that an angie inscribed in a semi-circle is a right angle

72 Let ABCD be a parallelogram Prove that AB2 + BC2 + CD2 + DA2 AC2 + BD2

73 If ABCD is any quadrilateral and P and Q are the midpoints of its diagonals prOVe that

AB2 + BC2 + CD2 + DA2 = AC2 + BD2 + 4PQ2 This is a generalization of the preceding problem

74 (a) Find an equation of a plane perpendicular to a given vector A and distant p from the origin (b) Express the eQuation of (a) in rectangular coordinates AilS (a) rmiddotn = p where D AA (b) A 1 x + ~y + Aaz = Ap

75 Let rl and r2 be unit vectors in the xy plane making angles a and 13 with the positive x-axis (a) Prove that r1 cos a 1 + sin a 1 r2 = cos f3 I + sin f3 j (b) By considering rl r2 prove the trigonometric formulas

cos (a - f3) = cos a cos f3 + sin a sin 13 cos (a +3) = cos a cosS - sin a sin S

71 Let a be the position vector of a given point (xlYl 1) and r the position vector of any point (xy ) Deshyscribe the locus ofrlf (a) Ir-al = 3 (b) (r-a)middota=O (e) (r-o)middotr=O AilS (a) Sphere centre at (Xl 11 1) and radius 3

(b) Plane perpendicular to a and passing through its terminal pOint (e) Sphere with centre at (x2 Y12 z2) and radius I x~ + Y~ + z~ or a sphere with a as diameter

77 Given that A = 31+j+21t and 8 1-2j-41t are the position vectors of points P and Q respectively (a) Find an eQuation for the plane passing through Q and perpendicular to line PQ (b) What is the distance from the point (-111) to the plane AilS (a) (r-8)(A-8) 0 or 2x+3y+6z -28 (b) 5

78 Evaluate each of the following (a) 2jx(31-4k) (b) (1+2j)xlt (e) (21-41t)x(I+2j) (d) (41+j-2k)x(31+1t) (e) (21+j-It)x(31-2j+4k) AilS (a)-81-6It (b) 21-j (e) 81-41+4k (d) I-I0j-3k (e) 21-11j-7k

79 If A =31-j-2k and B 21+3j+k find (a) IAxBI (b) (A+28)x(2A-8) (e) (A+8)x(A-B)I AilS (a) Yi95 (b) -251 +35j-5511 (e) 2ms

80 If A 1-2j-3k 8 21+j-1I and C = i+3j-2k find (e) I (A x8) x C I (e) A (8 xC) (e) (A xB) x (B xC) (b) IA x (B xC) I (d) (A xB)middot C (f) (A xB)(8middot C) AilS (a)5V2s (b31iO (e)-20 (d)-20 (e)-401-20j+201l laquof)351-35j+35k

81 Show that If A~ 0 and both of the conditions (a) A-B = AmiddotC and (b) Ax8 AxC hold simultaneously then B C but if 0nIy one of these conditions holds then 8 ~ C necessarily

82 Find the area of a parallelogram having diagonals A 31 +j - 2k and B = i - 3j + 4k AilS 5 v3

33 The DOT and CROSS PRODUCT

83 F1nd the area of a triangle with vertices at (3-12) 0-1-3) and (4-31) Ans i Ii65

84 It A = 21 + J - 3k and B 1- 2j +k find a vector of magnUude 5 perpendicular to both A and B

Ans plusmn 5f(I+J+k)

85 Use Problem 75 to derive the formulas s1n(ltl-t3) sinltlcost3 - cosltlslnt3 sin(ltl+t3) sinltlcost3 + cosltls1n~

86 A force given by F 31 + 2j - 4k is applied at the point (1 -12) Find the moment of F about the point (2 -I 3) Ans 21 - 1J - 2k

87 Tbe angular velocity of a rotating rigid body about an axis of rotation is given by Col 41 + J - 2k Find the linear velocity of a po1nt P on the body whose position vector relative to a point on the axis of rotation is 21-3J+k AlIs -51- 8J - 14k

88 Simplify (A+B)middot(B+C)x(C+A) Ans 2ABxC

Aa Ab Ac

89 Provethat (A BxC)(amiddotbxc) Bmiddota Bb Bmiddotc

Cmiddota Cmiddotb Cmiddotc

98 Find tbe volume of the parallelepiped whose edges are represented by A 21 - 3j +4k B =1 + 2J - k C 31- J + 2k Ans 7

91 It A B xC 0 show that either (0) A Band C are coplanar but no two of them are colllnear or (b) two of tbe vectors A B and C are colllnear or (c) all of tbe vectors A Band C are colllnear

92 Find tbe constant 0 such that the vectors 21-J + k 1 + 2J - 3t and 31 +aj + 5k are coplanar Ans 0 = -4

13 It A =a + 11b + z1C B 2a + 12b + Z2C and C sa + 1s b + sc prove that

1 11 1 ABxC 2 12 2 (amiddotbxc)

s 1s s

94 Prove that a necessary and sufficient condition tbat A x (B xC) (AxB)xC is (Ax C) x B = O Disshycuss the cases where A B 0 or Bmiddot C O

95 Let pOints P Q and R have position vectors r1 31 - 2j - k r2 i+ 3j +4k and r s = 21 + j - 2k relative to an origin O Find tbe distance from P to the plane OQR Ans 3

96 Find the shortest distance from (6-44) to the line joining (212) and (3-14) Ans 3

97 Given points P(213) Q(121) R(-1-2-2) and SO-40) find the shortest distance between lines PQ and RS Ans 3v2

98 Prove that the perpendiculars from the vertices of a triangle to the opposite sides (extended if necessary) meet in a point (theorthocentre of the triangle)

99 Prove that tbe perpendicular bisectors of the sides of a triangle meet in a point (the circumcentre of the trishyangle)

100 Prove that (AxB)middot(CxD) + (BxC)middot(AxD) + (CxA)middot(BxD) = o

101 Let PQR be a spherical triangle whose sides pqT are arcs of great circles Prove the law of cosirampe8 for sphericol triangles

cos p = cos q cos T + sin q sin r cos P

with analogous formulas for cos q and cos r obtained by cyclic permutation of tbe letters [Hlnt Interpret both aides of the identity (A xB) (A xC) (B C)(A A) - (A C)(B A)]

3If middot

COOOCR JEDOU DKCR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull Monday Ma bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullllABDALLMOHAMEDSEBA~G 71213181 13MAY bullbullbullbullbullbullbullbullbullbullbullbullbull~ bullbull 1~3 2013 112 AM 21MOHAMEDSOADKHALEDABHMAHMOUDABDALLAMR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull

~ 337 C 05JUN RUHC~~L~TIFMOHAMEDMRS E WEFILED3~~~ Di~UG CAIRUH H~~ ii~~ 0030 0 E SA VENDOR L EXISTS 1440 0 VENDOR R~CATOR DATA EXISTS gtFF SERVICE I~~DATA EXISTS gtVL CUSTOM CHECK TION EXISTS gtVR

NE-RUHTALT RULES EXIST gtSI

G~~IW966~~i~v~FG~~P REF CRI~RU OTAL 1033000

31- ( r -I

S =- ~--b J

J shy

~poundv fY)N 0 c ~(~-gt) l2

1

0

~ - t c -- ----shy

)t ) -- ----~

f-Y~[8

J

C-flt c f ~) J - -shy

~ I() f T ttr)~de A~ - _ e

-) f

f (~) f - fCb) ~ - ~ - 0- Ji~ - -J9~ )

---o J ~ ~

The DOT and CROSS PRODUCT 32

63 Find the projection of the vector 21 - 3j + 6k OD the vector 1 + 2j + 21t Ails 83

64 Find the projection of the vector 41 - 3j + k on the Une passing through the pOints (23-1) and (-2-43) AilS 1

85 If A 41 - j + 31t and 8 -21 + j - 21t find a unit vector perpendicular to both A and 8 AilS plusmn (I - 2j - 2k)3

61 Find the acute angle formed by two diagonals of a cUbe

67 Find a unit vector parallel to the xy plane and perpendicular to the vector 41- 3j +k AilS plusmn (31 +4j)5

68 Show that A =(21- 2j +1t)3 8 (I + 2j + 21t)3 and C = (21+ j - 2k)3 are mutually orthogonal unit vectors

69 Find the worlt done in moving an object along a straight line from (32-1) to (2-14) in a force field given by F = 41-3j+2k AilS 15

70 Let F be a constant vector force field Show that the work done in moving an object around any closed polshyygon in this force field is Zero

71 Prove that an angie inscribed in a semi-circle is a right angle

72 Let ABCD be a parallelogram Prove that AB2 + BC2 + CD2 + DA2 AC2 + BD2

73 If ABCD is any quadrilateral and P and Q are the midpoints of its diagonals prOVe that

AB2 + BC2 + CD2 + DA2 = AC2 + BD2 + 4PQ2 This is a generalization of the preceding problem

74 (a) Find an equation of a plane perpendicular to a given vector A and distant p from the origin (b) Express the eQuation of (a) in rectangular coordinates AilS (a) rmiddotn = p where D AA (b) A 1 x + ~y + Aaz = Ap

75 Let rl and r2 be unit vectors in the xy plane making angles a and 13 with the positive x-axis (a) Prove that r1 cos a 1 + sin a 1 r2 = cos f3 I + sin f3 j (b) By considering rl r2 prove the trigonometric formulas

cos (a - f3) = cos a cos f3 + sin a sin 13 cos (a +3) = cos a cosS - sin a sin S

71 Let a be the position vector of a given point (xlYl 1) and r the position vector of any point (xy ) Deshyscribe the locus ofrlf (a) Ir-al = 3 (b) (r-a)middota=O (e) (r-o)middotr=O AilS (a) Sphere centre at (Xl 11 1) and radius 3

(b) Plane perpendicular to a and passing through its terminal pOint (e) Sphere with centre at (x2 Y12 z2) and radius I x~ + Y~ + z~ or a sphere with a as diameter

77 Given that A = 31+j+21t and 8 1-2j-41t are the position vectors of points P and Q respectively (a) Find an eQuation for the plane passing through Q and perpendicular to line PQ (b) What is the distance from the point (-111) to the plane AilS (a) (r-8)(A-8) 0 or 2x+3y+6z -28 (b) 5

78 Evaluate each of the following (a) 2jx(31-4k) (b) (1+2j)xlt (e) (21-41t)x(I+2j) (d) (41+j-2k)x(31+1t) (e) (21+j-It)x(31-2j+4k) AilS (a)-81-6It (b) 21-j (e) 81-41+4k (d) I-I0j-3k (e) 21-11j-7k

79 If A =31-j-2k and B 21+3j+k find (a) IAxBI (b) (A+28)x(2A-8) (e) (A+8)x(A-B)I AilS (a) Yi95 (b) -251 +35j-5511 (e) 2ms

80 If A 1-2j-3k 8 21+j-1I and C = i+3j-2k find (e) I (A x8) x C I (e) A (8 xC) (e) (A xB) x (B xC) (b) IA x (B xC) I (d) (A xB)middot C (f) (A xB)(8middot C) AilS (a)5V2s (b31iO (e)-20 (d)-20 (e)-401-20j+201l laquof)351-35j+35k

81 Show that If A~ 0 and both of the conditions (a) A-B = AmiddotC and (b) Ax8 AxC hold simultaneously then B C but if 0nIy one of these conditions holds then 8 ~ C necessarily

82 Find the area of a parallelogram having diagonals A 31 +j - 2k and B = i - 3j + 4k AilS 5 v3

33 The DOT and CROSS PRODUCT

83 F1nd the area of a triangle with vertices at (3-12) 0-1-3) and (4-31) Ans i Ii65

84 It A = 21 + J - 3k and B 1- 2j +k find a vector of magnUude 5 perpendicular to both A and B

Ans plusmn 5f(I+J+k)

85 Use Problem 75 to derive the formulas s1n(ltl-t3) sinltlcost3 - cosltlslnt3 sin(ltl+t3) sinltlcost3 + cosltls1n~

86 A force given by F 31 + 2j - 4k is applied at the point (1 -12) Find the moment of F about the point (2 -I 3) Ans 21 - 1J - 2k

87 Tbe angular velocity of a rotating rigid body about an axis of rotation is given by Col 41 + J - 2k Find the linear velocity of a po1nt P on the body whose position vector relative to a point on the axis of rotation is 21-3J+k AlIs -51- 8J - 14k

88 Simplify (A+B)middot(B+C)x(C+A) Ans 2ABxC

Aa Ab Ac

89 Provethat (A BxC)(amiddotbxc) Bmiddota Bb Bmiddotc

Cmiddota Cmiddotb Cmiddotc

98 Find tbe volume of the parallelepiped whose edges are represented by A 21 - 3j +4k B =1 + 2J - k C 31- J + 2k Ans 7

91 It A B xC 0 show that either (0) A Band C are coplanar but no two of them are colllnear or (b) two of tbe vectors A B and C are colllnear or (c) all of tbe vectors A Band C are colllnear

92 Find tbe constant 0 such that the vectors 21-J + k 1 + 2J - 3t and 31 +aj + 5k are coplanar Ans 0 = -4

13 It A =a + 11b + z1C B 2a + 12b + Z2C and C sa + 1s b + sc prove that

1 11 1 ABxC 2 12 2 (amiddotbxc)

s 1s s

94 Prove that a necessary and sufficient condition tbat A x (B xC) (AxB)xC is (Ax C) x B = O Disshycuss the cases where A B 0 or Bmiddot C O

95 Let pOints P Q and R have position vectors r1 31 - 2j - k r2 i+ 3j +4k and r s = 21 + j - 2k relative to an origin O Find tbe distance from P to the plane OQR Ans 3

96 Find the shortest distance from (6-44) to the line joining (212) and (3-14) Ans 3

97 Given points P(213) Q(121) R(-1-2-2) and SO-40) find the shortest distance between lines PQ and RS Ans 3v2

98 Prove that the perpendiculars from the vertices of a triangle to the opposite sides (extended if necessary) meet in a point (theorthocentre of the triangle)

99 Prove that tbe perpendicular bisectors of the sides of a triangle meet in a point (the circumcentre of the trishyangle)

100 Prove that (AxB)middot(CxD) + (BxC)middot(AxD) + (CxA)middot(BxD) = o

101 Let PQR be a spherical triangle whose sides pqT are arcs of great circles Prove the law of cosirampe8 for sphericol triangles

cos p = cos q cos T + sin q sin r cos P

with analogous formulas for cos q and cos r obtained by cyclic permutation of tbe letters [Hlnt Interpret both aides of the identity (A xB) (A xC) (B C)(A A) - (A C)(B A)]

3If middot

COOOCR JEDOU DKCR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull Monday Ma bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullllABDALLMOHAMEDSEBA~G 71213181 13MAY bullbullbullbullbullbullbullbullbullbullbullbullbull~ bullbull 1~3 2013 112 AM 21MOHAMEDSOADKHALEDABHMAHMOUDABDALLAMR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull

~ 337 C 05JUN RUHC~~L~TIFMOHAMEDMRS E WEFILED3~~~ Di~UG CAIRUH H~~ ii~~ 0030 0 E SA VENDOR L EXISTS 1440 0 VENDOR R~CATOR DATA EXISTS gtFF SERVICE I~~DATA EXISTS gtVL CUSTOM CHECK TION EXISTS gtVR

NE-RUHTALT RULES EXIST gtSI

G~~IW966~~i~v~FG~~P REF CRI~RU OTAL 1033000

31- ( r -I

S =- ~--b J

J shy

~poundv fY)N 0 c ~(~-gt) l2

1

0

~ - t c -- ----shy

)t ) -- ----~

f-Y~[8

The DOT and CROSS PRODUCT 32

63 Find the projection of the vector 21 - 3j + 6k OD the vector 1 + 2j + 21t Ails 83

64 Find the projection of the vector 41 - 3j + k on the Une passing through the pOints (23-1) and (-2-43) AilS 1

85 If A 41 - j + 31t and 8 -21 + j - 21t find a unit vector perpendicular to both A and 8 AilS plusmn (I - 2j - 2k)3

61 Find the acute angle formed by two diagonals of a cUbe

67 Find a unit vector parallel to the xy plane and perpendicular to the vector 41- 3j +k AilS plusmn (31 +4j)5

68 Show that A =(21- 2j +1t)3 8 (I + 2j + 21t)3 and C = (21+ j - 2k)3 are mutually orthogonal unit vectors

69 Find the worlt done in moving an object along a straight line from (32-1) to (2-14) in a force field given by F = 41-3j+2k AilS 15

70 Let F be a constant vector force field Show that the work done in moving an object around any closed polshyygon in this force field is Zero

71 Prove that an angie inscribed in a semi-circle is a right angle

72 Let ABCD be a parallelogram Prove that AB2 + BC2 + CD2 + DA2 AC2 + BD2

73 If ABCD is any quadrilateral and P and Q are the midpoints of its diagonals prOVe that

AB2 + BC2 + CD2 + DA2 = AC2 + BD2 + 4PQ2 This is a generalization of the preceding problem

74 (a) Find an equation of a plane perpendicular to a given vector A and distant p from the origin (b) Express the eQuation of (a) in rectangular coordinates AilS (a) rmiddotn = p where D AA (b) A 1 x + ~y + Aaz = Ap

75 Let rl and r2 be unit vectors in the xy plane making angles a and 13 with the positive x-axis (a) Prove that r1 cos a 1 + sin a 1 r2 = cos f3 I + sin f3 j (b) By considering rl r2 prove the trigonometric formulas

cos (a - f3) = cos a cos f3 + sin a sin 13 cos (a +3) = cos a cosS - sin a sin S

71 Let a be the position vector of a given point (xlYl 1) and r the position vector of any point (xy ) Deshyscribe the locus ofrlf (a) Ir-al = 3 (b) (r-a)middota=O (e) (r-o)middotr=O AilS (a) Sphere centre at (Xl 11 1) and radius 3

(b) Plane perpendicular to a and passing through its terminal pOint (e) Sphere with centre at (x2 Y12 z2) and radius I x~ + Y~ + z~ or a sphere with a as diameter

77 Given that A = 31+j+21t and 8 1-2j-41t are the position vectors of points P and Q respectively (a) Find an eQuation for the plane passing through Q and perpendicular to line PQ (b) What is the distance from the point (-111) to the plane AilS (a) (r-8)(A-8) 0 or 2x+3y+6z -28 (b) 5

78 Evaluate each of the following (a) 2jx(31-4k) (b) (1+2j)xlt (e) (21-41t)x(I+2j) (d) (41+j-2k)x(31+1t) (e) (21+j-It)x(31-2j+4k) AilS (a)-81-6It (b) 21-j (e) 81-41+4k (d) I-I0j-3k (e) 21-11j-7k

79 If A =31-j-2k and B 21+3j+k find (a) IAxBI (b) (A+28)x(2A-8) (e) (A+8)x(A-B)I AilS (a) Yi95 (b) -251 +35j-5511 (e) 2ms

80 If A 1-2j-3k 8 21+j-1I and C = i+3j-2k find (e) I (A x8) x C I (e) A (8 xC) (e) (A xB) x (B xC) (b) IA x (B xC) I (d) (A xB)middot C (f) (A xB)(8middot C) AilS (a)5V2s (b31iO (e)-20 (d)-20 (e)-401-20j+201l laquof)351-35j+35k

81 Show that If A~ 0 and both of the conditions (a) A-B = AmiddotC and (b) Ax8 AxC hold simultaneously then B C but if 0nIy one of these conditions holds then 8 ~ C necessarily

82 Find the area of a parallelogram having diagonals A 31 +j - 2k and B = i - 3j + 4k AilS 5 v3

33 The DOT and CROSS PRODUCT

83 F1nd the area of a triangle with vertices at (3-12) 0-1-3) and (4-31) Ans i Ii65

84 It A = 21 + J - 3k and B 1- 2j +k find a vector of magnUude 5 perpendicular to both A and B

Ans plusmn 5f(I+J+k)

85 Use Problem 75 to derive the formulas s1n(ltl-t3) sinltlcost3 - cosltlslnt3 sin(ltl+t3) sinltlcost3 + cosltls1n~

86 A force given by F 31 + 2j - 4k is applied at the point (1 -12) Find the moment of F about the point (2 -I 3) Ans 21 - 1J - 2k

87 Tbe angular velocity of a rotating rigid body about an axis of rotation is given by Col 41 + J - 2k Find the linear velocity of a po1nt P on the body whose position vector relative to a point on the axis of rotation is 21-3J+k AlIs -51- 8J - 14k

88 Simplify (A+B)middot(B+C)x(C+A) Ans 2ABxC

Aa Ab Ac

89 Provethat (A BxC)(amiddotbxc) Bmiddota Bb Bmiddotc

Cmiddota Cmiddotb Cmiddotc

98 Find tbe volume of the parallelepiped whose edges are represented by A 21 - 3j +4k B =1 + 2J - k C 31- J + 2k Ans 7

91 It A B xC 0 show that either (0) A Band C are coplanar but no two of them are colllnear or (b) two of tbe vectors A B and C are colllnear or (c) all of tbe vectors A Band C are colllnear

92 Find tbe constant 0 such that the vectors 21-J + k 1 + 2J - 3t and 31 +aj + 5k are coplanar Ans 0 = -4

13 It A =a + 11b + z1C B 2a + 12b + Z2C and C sa + 1s b + sc prove that

1 11 1 ABxC 2 12 2 (amiddotbxc)

s 1s s

94 Prove that a necessary and sufficient condition tbat A x (B xC) (AxB)xC is (Ax C) x B = O Disshycuss the cases where A B 0 or Bmiddot C O

95 Let pOints P Q and R have position vectors r1 31 - 2j - k r2 i+ 3j +4k and r s = 21 + j - 2k relative to an origin O Find tbe distance from P to the plane OQR Ans 3

96 Find the shortest distance from (6-44) to the line joining (212) and (3-14) Ans 3

97 Given points P(213) Q(121) R(-1-2-2) and SO-40) find the shortest distance between lines PQ and RS Ans 3v2

98 Prove that the perpendiculars from the vertices of a triangle to the opposite sides (extended if necessary) meet in a point (theorthocentre of the triangle)

99 Prove that tbe perpendicular bisectors of the sides of a triangle meet in a point (the circumcentre of the trishyangle)

100 Prove that (AxB)middot(CxD) + (BxC)middot(AxD) + (CxA)middot(BxD) = o

101 Let PQR be a spherical triangle whose sides pqT are arcs of great circles Prove the law of cosirampe8 for sphericol triangles

cos p = cos q cos T + sin q sin r cos P

with analogous formulas for cos q and cos r obtained by cyclic permutation of tbe letters [Hlnt Interpret both aides of the identity (A xB) (A xC) (B C)(A A) - (A C)(B A)]

3If middot

COOOCR JEDOU DKCR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull Monday Ma bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullllABDALLMOHAMEDSEBA~G 71213181 13MAY bullbullbullbullbullbullbullbullbullbullbullbullbull~ bullbull 1~3 2013 112 AM 21MOHAMEDSOADKHALEDABHMAHMOUDABDALLAMR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull

~ 337 C 05JUN RUHC~~L~TIFMOHAMEDMRS E WEFILED3~~~ Di~UG CAIRUH H~~ ii~~ 0030 0 E SA VENDOR L EXISTS 1440 0 VENDOR R~CATOR DATA EXISTS gtFF SERVICE I~~DATA EXISTS gtVL CUSTOM CHECK TION EXISTS gtVR

NE-RUHTALT RULES EXIST gtSI

G~~IW966~~i~v~FG~~P REF CRI~RU OTAL 1033000

31- ( r -I

S =- ~--b J

J shy

~poundv fY)N 0 c ~(~-gt) l2

1

0

~ - t c -- ----shy

)t ) -- ----~

f-Y~[8

33 The DOT and CROSS PRODUCT

83 F1nd the area of a triangle with vertices at (3-12) 0-1-3) and (4-31) Ans i Ii65

84 It A = 21 + J - 3k and B 1- 2j +k find a vector of magnUude 5 perpendicular to both A and B

Ans plusmn 5f(I+J+k)

85 Use Problem 75 to derive the formulas s1n(ltl-t3) sinltlcost3 - cosltlslnt3 sin(ltl+t3) sinltlcost3 + cosltls1n~

86 A force given by F 31 + 2j - 4k is applied at the point (1 -12) Find the moment of F about the point (2 -I 3) Ans 21 - 1J - 2k

87 Tbe angular velocity of a rotating rigid body about an axis of rotation is given by Col 41 + J - 2k Find the linear velocity of a po1nt P on the body whose position vector relative to a point on the axis of rotation is 21-3J+k AlIs -51- 8J - 14k

88 Simplify (A+B)middot(B+C)x(C+A) Ans 2ABxC

Aa Ab Ac

89 Provethat (A BxC)(amiddotbxc) Bmiddota Bb Bmiddotc

Cmiddota Cmiddotb Cmiddotc

98 Find tbe volume of the parallelepiped whose edges are represented by A 21 - 3j +4k B =1 + 2J - k C 31- J + 2k Ans 7

91 It A B xC 0 show that either (0) A Band C are coplanar but no two of them are colllnear or (b) two of tbe vectors A B and C are colllnear or (c) all of tbe vectors A Band C are colllnear

92 Find tbe constant 0 such that the vectors 21-J + k 1 + 2J - 3t and 31 +aj + 5k are coplanar Ans 0 = -4

13 It A =a + 11b + z1C B 2a + 12b + Z2C and C sa + 1s b + sc prove that

1 11 1 ABxC 2 12 2 (amiddotbxc)

s 1s s

94 Prove that a necessary and sufficient condition tbat A x (B xC) (AxB)xC is (Ax C) x B = O Disshycuss the cases where A B 0 or Bmiddot C O

95 Let pOints P Q and R have position vectors r1 31 - 2j - k r2 i+ 3j +4k and r s = 21 + j - 2k relative to an origin O Find tbe distance from P to the plane OQR Ans 3

96 Find the shortest distance from (6-44) to the line joining (212) and (3-14) Ans 3

97 Given points P(213) Q(121) R(-1-2-2) and SO-40) find the shortest distance between lines PQ and RS Ans 3v2

98 Prove that the perpendiculars from the vertices of a triangle to the opposite sides (extended if necessary) meet in a point (theorthocentre of the triangle)

99 Prove that tbe perpendicular bisectors of the sides of a triangle meet in a point (the circumcentre of the trishyangle)

100 Prove that (AxB)middot(CxD) + (BxC)middot(AxD) + (CxA)middot(BxD) = o

101 Let PQR be a spherical triangle whose sides pqT are arcs of great circles Prove the law of cosirampe8 for sphericol triangles

cos p = cos q cos T + sin q sin r cos P

with analogous formulas for cos q and cos r obtained by cyclic permutation of tbe letters [Hlnt Interpret both aides of the identity (A xB) (A xC) (B C)(A A) - (A C)(B A)]

3If middot

COOOCR JEDOU DKCR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull Monday Ma bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullllABDALLMOHAMEDSEBA~G 71213181 13MAY bullbullbullbullbullbullbullbullbullbullbullbullbull~ bullbull 1~3 2013 112 AM 21MOHAMEDSOADKHALEDABHMAHMOUDABDALLAMR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull

~ 337 C 05JUN RUHC~~L~TIFMOHAMEDMRS E WEFILED3~~~ Di~UG CAIRUH H~~ ii~~ 0030 0 E SA VENDOR L EXISTS 1440 0 VENDOR R~CATOR DATA EXISTS gtFF SERVICE I~~DATA EXISTS gtVL CUSTOM CHECK TION EXISTS gtVR

NE-RUHTALT RULES EXIST gtSI

G~~IW966~~i~v~FG~~P REF CRI~RU OTAL 1033000

31- ( r -I

S =- ~--b J

J shy

~poundv fY)N 0 c ~(~-gt) l2

1

0

~ - t c -- ----shy

)t ) -- ----~

f-Y~[8

3If middot

COOOCR JEDOU DKCR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull Monday Ma bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullllABDALLMOHAMEDSEBA~G 71213181 13MAY bullbullbullbullbullbullbullbullbullbullbullbullbull~ bullbull 1~3 2013 112 AM 21MOHAMEDSOADKHALEDABHMAHMOUDABDALLAMR bullbullbullbullbullbullbullbullbullbullbullbullbullbullbull

~ 337 C 05JUN RUHC~~L~TIFMOHAMEDMRS E WEFILED3~~~ Di~UG CAIRUH H~~ ii~~ 0030 0 E SA VENDOR L EXISTS 1440 0 VENDOR R~CATOR DATA EXISTS gtFF SERVICE I~~DATA EXISTS gtVL CUSTOM CHECK TION EXISTS gtVR

NE-RUHTALT RULES EXIST gtSI

G~~IW966~~i~v~FG~~P REF CRI~RU OTAL 1033000

31- ( r -I

S =- ~--b J

J shy

~poundv fY)N 0 c ~(~-gt) l2

1

0

~ - t c -- ----shy

)t ) -- ----~

f-Y~[8

I

~5

-- - -

---

2) C[ f- 11) tgt ~j~(

) Yj __ )7 I --==~--~~

r shy( ) tJ rJ~-~ 7 J-r ~~k15- ( 7 7 - 11 cI 1 ~ J -t 1

~ ~ olT) 1- 5 r Cshy

-i-~7t ~ r Ei 2 X - J l ~e t) ) 7- L

y~ I) if -_~l-7-l1 ---- -

L

ff~ [j fl [l r -= - J c t - j ~ f ltmiddotf J + 5 _ J i Oicdl)

- 4- (X +2) - it 0 -2) ~ ~ (~ - t) [ J k - - )(+L +J- 1 -f i-I

Jlt ~ 2- - _j C) t ( ~t- 1 ltJt ) [ - lt -I ( C1( - 2Emiddot f~)ikJ~t (IY ) 1 7) f shy---t ~ _ 3 c5 t (I - gts 1 ) ~ - 3 ~ f (- 5cf ~

lrlI~ __ -----------)----------h r~ 1 __ lSt

shy

---~------

l 1 shyy - _ ) lt) f c i t (e~ t - S t ) )

Distance from point to line - 3-Dimensional Page 1 of2

About I Practice I Calculators I Feedback Englllll i

About Distance from point to line - 3-Dimensional News Distance from point to line is equal to length of the

perpendicular distance from the point to the line Practice

If Mo(xo yo zo) pOint coordinates i = 1m n p - directing Calculators vector of line I Ml(Xl Yl Zl) - coordinates of point on line I

then distance between point Mo(xo yo zo) and line I can beLibrary

found using the following formula 1oFormulas

Feedback d= ___

Iii

Formula proving If I is line equation then i = 1m n p) - directing vector of line Ml(X1 1 Yl Zl) - coordinates of point on line

From properties of cross product it is known that the module of cross product of vectors is equal to the area of a parallelogramme constructed on these vectors

S = IMoMlshy

On the other hand parallelogramme area is equal to product of its side on height spent to this side S laid

Having equated the areas it is simple to receive the formula of distance from a point to a line

Example To find distance between point M(O 2 3) and line

x-3 y1 z+1 --=--=-shy

2 2

Solution From line equation find a= 2 1 2 - directing vector of line

MI(3 1 -1) - coordinates of point on line

Then

~ = (3 - 0 1 - 2 -1 - 3) = 3 -1 -4

i j k I~a= ~ -1 _ = I =i laquo-1)-2 - (-4)1) - j (3middot2 - (-4)middot2) + k (H -(-1)2) =2 -14 51

225 112I~al (22 + (-14)2 + 52)12 d=--_ = --=5

lal (22 + 12 + 22)112 912

Answer distance from point to line is equal to 5

Distance Learning Courses wwwstaffordaeDistantLearn

Internationally Recognized Degrees Unlock Your Potential Online Today

See alsGO

Online calculator [lstance from point to line - 3-Dlmenslonal

Analyllc geo_by bull Distance from Int to plane bull Distance from ~ane to plane bull Distance from point to line - Z-Dlmenslonal

Onlij

322013httponlinemschoolcomlmathllibrary lanalytic ~eometryIpJinel

Distance from point to line - 3-Dimensional Page 1 of2

About I Practice I Calculators I Feedback Englllll i

About Distance from point to line - 3-Dimensional News Distance from point to line is equal to length of the

perpendicular distance from the point to the line Practice

If Mo(xo yo zo) pOint coordinates i = 1m n p - directing Calculators vector of line I Ml(Xl Yl Zl) - coordinates of point on line I

then distance between point Mo(xo yo zo) and line I can beLibrary

found using the following formula 1oFormulas

Feedback d= ___

Iii

Formula proving If I is line equation then i = 1m n p) - directing vector of line Ml(X1 1 Yl Zl) - coordinates of point on line

From properties of cross product it is known that the module of cross product of vectors is equal to the area of a parallelogramme constructed on these vectors

S = IMoMlshy

On the other hand parallelogramme area is equal to product of its side on height spent to this side S laid

Having equated the areas it is simple to receive the formula of distance from a point to a line

Example To find distance between point M(O 2 3) and line

x-3 y1 z+1 --=--=-shy

2 2

Solution From line equation find a= 2 1 2 - directing vector of line

MI(3 1 -1) - coordinates of point on line

Then

~ = (3 - 0 1 - 2 -1 - 3) = 3 -1 -4

i j k I~a= ~ -1 _ = I =i laquo-1)-2 - (-4)1) - j (3middot2 - (-4)middot2) + k (H -(-1)2) =2 -14 51

225 112I~al (22 + (-14)2 + 52)12 d=--_ = --=5

lal (22 + 12 + 22)112 912

Answer distance from point to line is equal to 5

Distance Learning Courses wwwstaffordaeDistantLearn

Internationally Recognized Degrees Unlock Your Potential Online Today

See alsGO

Online calculator [lstance from point to line - 3-Dlmenslonal

Analyllc geo_by bull Distance from Int to plane bull Distance from ~ane to plane bull Distance from point to line - Z-Dlmenslonal

Onlij

322013httponlinemschoolcomlmathllibrary lanalytic ~eometryIpJinel

Point-Line Distance--3-Dimensional-- from Wolfram MathWorld

Search MathWorld Yl)ltia nl MathWrld

Algebra

Applied Mathematics Geometrygt Distancegt Geometrygt Line Geometrygt Lmes gt

Calculus and Analysis Interactive Enlnes gt Interactive DemonstratIOnsgt

Disaete Mathematics THINGS TO TRY

Foundations of Mathematics Point-Line Distance--3-Dimensional polnHine distance-~

MatnKtlrld Classroom Let aline in three dimensions be specifted by two points 11 =(r bull 1 bull t I ) and = (X bull n l lying on it so 8 calculus homework vador along the line is given by About MatnvtJr1d done faster

Contribute to MathWorid IXI +~lt -xd I (1)

Send a Message to the Team V= I Y + () middotYI)t

~ (r~ 1 The squered distence beloveen e point on the line with perarreter end e point lt - ( Y lt) is therafore

Wolfram Web ResQwces bull

Matnw)rld Book

(2)

1313genlfles Last updated Tue Feb 26 2013 To mininize tha distance set d IJId t = Co and solve for to obtain

= (Ie X)+ fn - x )f +1( -~oi +C y)t1 +(u ltoJ +(n - dtl2

Created tkVftO~d and nurtured by Eric Weirrlein (II lIo) J[ middot1tJ (3) at Wolfr~ Reampet1lCIr Illl -xI

where denotes the dot product Tha minimum distance can then be found by plugging back int) (2) to obtain

(4)1=1-1 _~l +() yof +(1 middotuf+ 2 [tn ) x - 0)+ y - y)(y -yo) +(z bullbull ( zoll bull

2 [( lt r j (y d j ( lt21 Illi --loH lll)l (~I - Jo) (lll llln (5)-Is 11)12 2 j

11 bull ~I~~~~ii = IgtL~~~~ I(s~)~( Ill] (6)

III II I

Using the vector quadruple produd

(A)(Br=AB -(AS) (7)

whare)( denotes the cross produclthen giws

Geometry

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dimensional

105middot12000

diletragonal pyramidal

1nt~fKthwk~PfIlm

~i~ l lt ~ ~I

Constructing Vedor Georretry Solution Michael Rogers

Gt1 your

(8)

and taking tha square root results in the beautiful formula

~ - I(x X )(x )1 (9) -~p-~~~--~-

II middotllli = 1(10 bull II ))((Xo bull 121 (10)

Ill -ll1 (11)

Here the numerat)r is simpty twice the area of tha triangla fonned by points ~t II and It and the denominabr is the lengtl of one of the bases of tha lrIangle which follOWS since from the ullal triangle area formula ~= b d1

SEE ALSO Collinear Line Point Point-Line Distance--2 p Dimensional Triangle Area

CITE nilS AS vJeisstein Eric W Polnt-Une Distance-~Dlmensk)nalmiddot Frttm Math World-A M)lfram Neb Resource hplmathworldwotrramcomPointpLineDislance~Dimensionalhtml

Wolfram Web Resources

Mathematlcaraquo WoWramlAlpha raquo WOlfram Demonstrations Project ) The 1 t)ol for a-eatlng Explore an~hing with the first Explore thousands of free applicatons Demonstations and anything computational knowedge engine across science mathematics technical engineering tedlOology business an

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httpmathworld wolframcomiP oint -LineDistance3 -Dimensionalhtml 322013

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~ X)( ~ J -+- 1 C -= 9 x -+ r tl1 c-P0

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c

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c~ (-x- oJ -- (r l )(-4-) - ~13 Li-) c

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Ij(f-) -7f1-tr)-)~ = ~

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t v

G)-)j-t~ -L

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bull

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L

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bull I bull

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c

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CCl-X) _ crfr(j-~ )_5vJt = c

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t v

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~ V3~ -trJ+~-t tZ

bull

= ~ -r _l )1_ 00 +I (Jl-

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(If t J~ (~l + ~) L J~ ~1 + -- t b~+ h-) I) ) ( L1--rshy- ~ 1gt 2~1~ncL~- - (~1~1)

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bull I bull

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1 1 r

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I- - ~ -+- ~ l~ -+- ~~)

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VAl 1 jf -r tzj b1 0 C~ + t ~)I

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1 1 r

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( ~ UoLf C A ) ( a(J )

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