Vectors and 3-D Geometry - arbindsingh.com |...
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LOCUS LOCUS LOCUS LOCUS LOCUS 18 Mathematics / Vectors and 3-D Geometry RESOLUTION OF A VECTOR IN A GIVEN BASIS a b r a b a b Fig - 21 r b B O A a r OA OB a b OA OB r a b r a b OA OB RECTANGULAR RESOLUTION i j Fig - 22 r B A j ^ O i ^ r i ^ OA r OB r
Transcript of Vectors and 3-D Geometry - arbindsingh.com |...
LOCUSLOCUSLOCUSLOCUSLOCUS 18
Mathematics / Vectors and 3-D Geometry
RESOLUTION OF A VECTOR IN A GIVEN BASIS
a b
r a b a b
Fig - 21
rb
B
O Aa
r OA OB
a b
OA OB r a br a b
OA OB
RECTANGULAR RESOLUTION
i j
Fig - 22
r
B
A
j
O i
r i
OA r
OB r
LOCUSLOCUSLOCUSLOCUSLOCUS 19
Mathematics / Vectors and 3-D Geometry
r
r OA OB
r i r j
xi yj
x y x y r
i j k
Fig - 23
r
j
i
k
O
r x y z
r xi yj zk
i j i j k
LOCUSLOCUSLOCUSLOCUSLOCUS 20
Mathematics / Vectors and 3-D Geometry
MAGNITUDE, DIRECTION COSINES AND DIRECTION RATIOS
r xi yj zk
Fig - 24
r
y
xz
Z
X
Y
r x y z
r x y z
r X Y Z
Fig - 25
r
Z
X
Y
l
m
n
direction cosines r DC DC
r xi yj zk
LOCUSLOCUSLOCUSLOCUSLOCUS 21
Mathematics / Vectors and 3-D Geometry
x r l r
y r m r
z r n r
x y z l m n r
l m n
r r
r xi yj zkrr r
li yj zk
DR a b cDC
l m na b c
DR DC
DR a b c DC
l m n l m na b c a b c a b c
a b cl m na b c a b c a b c
i j k i k k i j k
Example – 6
LOCUSLOCUSLOCUSLOCUSLOCUS 22
Mathematics / Vectors and 3-D Geometry
Solution:
i j k i j k i j k
i j k
a b c A a b c B a b c
C a b c P p
O OP p
Solution:
A B C
O
Fig - 26
A BC
OA, OB OC
BC AB
Example – 7 COLLINEARITY OF POINTS
LOCUSLOCUSLOCUSLOCUSLOCUS 23
Mathematics / Vectors and 3-D Geometry
BC AB
OC OB OB OA
OA OB OC
a b c a b c a b c
a b c
a b c
A B C
a b c A a b c B a b c
C a b c D a b c
Solution:
Fig - 27
B
A C
D
AB AD
C
ACAB AD AC
AB AD
Example – 8
LOCUSLOCUSLOCUSLOCUSLOCUS 24
Mathematics / Vectors and 3-D Geometry
AC AB AD
OC OA OB OA OD OA O
a b c a b c a b c
a b c
a b c
AB AC AD
A B C D
A a B b
m n
Solution:
C c AB m n
A
B
OFig - 28
Cm : n
ac b
Example – 9 SECTION FORMULA
LOCUSLOCUSLOCUSLOCUSLOCUS 25
Mathematics / Vectors and 3-D Geometry
mAC ABm n
mOC OA OB OAm n
mc a b am n
mb nacm n
D d AB m n
mb nadm n
A a B ba b
Solution: A a B b C c
Fig - 29
yz
I
D C cB b
A a
x
x y z
A BI
Example – 10
LOCUSLOCUSLOCUSLOCUSLOCUS 26
Mathematics / Vectors and 3-D Geometry
BD cDC b
D
zc ybDz y
ABD BI
z xDI BD xz yIA BA z z y
A D I AD I
zc ybxa x yz y
Ix z y
xa yb zcx y z
C II
ABCD M AB AC MD E AC MDE
Solution:
CD
A BM
E
AEEC = ME
ED =
Fig - 30
Example – 11
LOCUSLOCUSLOCUSLOCUSLOCUS 27
Mathematics / Vectors and 3-D Geometry
A B C
b c D AD
AD AC CD
c AB
c bD c b
M AB M' b
E' AC DMDM
c cE
E MD
b c b cE
E E E AC MD E
Solution:
Fig - 31
EFG
D C cB b
A a
AGGD =
Example – 12
LOCUSLOCUSLOCUSLOCUSLOCUS 28
Mathematics / Vectors and 3-D Geometry
D BC
b cD
AG GD G
b c aa b cG
A a B b C c D d
Fig - 32
C c
A a
D d
GBCD
G
B b
BCD G
b c dG
AG
P AGP
b c d aa b c dP
PP
LOCUSLOCUSLOCUSLOCUSLOCUS 29
Mathematics / Vectors and 3-D Geometry
PQRS PQ a QR b SP a b M QR X SMSX SM P X R
Solution:
P
M
R a + b
Q aP O
S b – a
XX
SX SM
SX XM
Fig - 33
a a b bM a
ba b aa bX
X a b
PX PR
P X R
Example – 13
LOCUSLOCUSLOCUSLOCUSLOCUS 30
Mathematics / Vectors and 3-D Geometry
ABC G O H
OG GH
P ABC
(a) OG OB OC OH (b) HA HB HC HO (c) PA PB PC PG
Solution:
Fig - 34
DE
O
F BA
C
D E F
BC CA ABGH
(a) OA OB OC OA OB OC
OA OD D BC'
OG G AD
OH O G H OH OG
(b) HA HB HC HA HB HC
HA HD
HG
HO
HO(c) P ABC
PA PB PC PA PB PC
PA PD
PG
Example – 14
LOCUSLOCUSLOCUSLOCUSLOCUS 31
Mathematics / Vectors and 3-D Geometry
(a) a b c p q r
pa qb rc
p q r
(b) a b c d p q r s
pa qb rc sd
p q r s
Solution:
(a) p q r r p q
r
pa qb rc
pa qb p q c
pa qbcp q
c a b
p q a b c
a b c
b a c a
a b c
pa qb rc
p q r
Example – 15
LOCUSLOCUSLOCUSLOCUSLOCUS 32
Mathematics / Vectors and 3-D Geometry
(b) p q r s
p q r s
s p q r
pa qb rc sd
pa qb rc p q r d
p a d q b d r c d
p
q ra d b d c dp p
b d c d
a d b d c d
a d b d c d
a b c d
A a B b C c D d
AB AC AD
b a c a d a
a b c d
pa qb rc sd
p q r s
LOCUSLOCUSLOCUSLOCUSLOCUS 33
Mathematics / Vectors and 3-D Geometry
O ABCD AO BO CO DOP Q R S
OP OQ OR OSAP BQ CR DS
Solution: O A B C D a b c d
a b c d
a b c d
AO P AO OP
AO OP
a OPp p P
p b c d
B C D P
OPOPAP AP
i
i i i
OQ OR OSBQ CR DS
OP OQ OR OSAP BQ CR DS
Example – 16
LOCUSLOCUSLOCUSLOCUSLOCUS 34
Mathematics / Vectors and 3-D Geometry
Q. 1 i j i j ai j a
Q. 2 a b c a b c
a b c b c
Q. 3 P ABCD
PA PB PC PD
Q. 4 ABCD L M BC CD
AL AM AC
Q. 5
Q. 6
Q. 7 ABCD E F AC BD
AB AD CB CD EF
Q. 8 a b c a b c a b c
a b c
Q. 9 a b c ma b c
a mb c a b mc m
Q. 10 ABC D AC AD DC BD F DF BDAF BC BD
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