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LOCUSLOCUSLOCUSLOCUSLOCUS 1
Mathematics / Vectors and 3-D Geometry
3
01. Introduction
02. Basic Vector Operations
03. Dot Product
04. Product Product
05. Scalar Triple Product
06. Vector Triple Product
07. More Geometry with Vectors
08. Appendix : 3-D Geometry
CONCEPT NOCONCEPT NOCONCEPT NOCONCEPT NOCONCEPT NOTESTESTESTESTES
LOCUSLOCUSLOCUSLOCUSLOCUS 2
Mathematics / Vectors and 3-D Geometry
3
vectorA B
B
AFig - 1
AB
(a)
(b)
A B
scalar
F
M
Fig - 2
N
Section - 1 INTRODUCTION
LOCUSLOCUSLOCUSLOCUSLOCUS 3
Mathematics / Vectors and 3-D Geometry
AB aa
Length : a a
Support : a
Sense : PQ P Q PQ QP
Q P PQ
(A) Zero vector :
(B) Unit vectors: a a aa
(C) Collinear vectors:
O A
OA O
aa
LOCUSLOCUSLOCUSLOCUSLOCUS 4
Mathematics / Vectors and 3-D Geometry
(D) Equal vectors : a b
a b
(E) Co-initial vectors: Fixed
(F) Co-terminus Fixed vectors:
(G) Co-planar vectors: free
a b c a b
O a b cO c
a b a b c
b
O
c
a
a b
Fig - 3
c a b
bc
aO
c a b
(H) Negative of a : a a a vector
(I) Position vector : PP
LOCUSLOCUSLOCUSLOCUSLOCUS 5
Mathematics / Vectors and 3-D Geometry
(A) ADDITION OF VECTORS : TRIANGLE / PARALLELOGRAM LAW
a b
c a b
c a b a b
b a
b
Fig - 4
B
C
Aa
A a b C
a b A C a b c AC
Fig - 5
b
B
C
Aa
c c a b= +AC=
Section - 2 BASIC VECTOR OPERATIONS
LOCUSLOCUSLOCUSLOCUSLOCUS 6
Mathematics / Vectors and 3-D Geometry
a b b
a a b a
b triangle law of vector addition a b
Fig - 6
b
A
C
O a
BC a=b=
B
AC
OC a b
Fig - 7
b
A
C
O a
B
ba +
(a) Existence of identity: a
a a
(b) Existence of inverse: a
a a
(c) Commutativity: a b
a b b a
LOCUSLOCUSLOCUSLOCUSLOCUS 7
Mathematics / Vectors and 3-D Geometry
(d) Associativity: a b c
a b c a b c
(B) SUBTRACTION OF VECTORS : An extension of addition
a b c
c a b
c a b
b b
a b
bb
a
a
b
bb
a
b –b
a -b a – b OR
a b
b a a – b
Fig - 8
b a a b a b
LOCUSLOCUSLOCUSLOCUSLOCUS 8
Mathematics / Vectors and 3-D Geometry
a b c
c
a
Fig - 9
b
a b c
ia i n n
a
a
aaan
Fig - 10
na a a
a b
a b a b a b a b a b a b
Example – 1
LOCUSLOCUSLOCUSLOCUSLOCUS 9
Mathematics / Vectors and 3-D Geometry
Solution:
Fig - 11
b
B
C
Aa
– b
ba
ba
C'
In : AC AB BC
a b a b
In ': AC AB BC AB BC
a b a b
aA
bBO
Fig - 12
|a + b| = OB
= OA + AB
= | a | + | b |
a AB'
O
Fig - 13
–bbB
A
|a – b| = |a + –b
= |OB' |
= |OA + AB'|
= | a | + | b |
LOCUSLOCUSLOCUSLOCUSLOCUS 10
Mathematics / Vectors and 3-D Geometry
ABC
AB BC AC
a b a b
a b
a AO
Fig - 14
bB A
|a + b| = |OA + AB|
= |OB|
= |OA – AB|
= || a | – | b ||
B
a b
Solution: A A A A A A
A
AA
A
A A
Fig - 15
a
b
A A A A
A A A A
A A A A
b
Example – 2
LOCUSLOCUSLOCUSLOCUSLOCUS 11
Mathematics / Vectors and 3-D Geometry
A A A A A A
b a b
b a
A A aa
A A A A
b
A A A A
a b
a b
a b
a b a b
Solution: a b
Fig - 16
b
A
C
O a
B
a b OC OC
a b BA BA
OACBOACB
a b a b
Example – 3
LOCUSLOCUSLOCUSLOCUSLOCUS 12
Mathematics / Vectors and 3-D Geometry
(C) MULTIPLICATION OF A VECTOR BY A SCALAR
a
Fig - 17
a
a
>
a
a
<
a a a
a a a
a
aaa
a b
a b Collinear vectors
non a b
a b
a b
LOCUSLOCUSLOCUSLOCUSLOCUS 13
Mathematics / Vectors and 3-D Geometry
a b a b
a b
linear combinationsn na a a n
r
n nr a a a
n
a b
THE BASIS OF A VECTOR SPACE
a b
Fig - 18
b
a
any r a b components
r a b a b
Fig - 19
b
a
B C
A
r
r = OC
= OA + OB
= a + b
O
LOCUSLOCUSLOCUSLOCUSLOCUS 14
Mathematics / Vectors and 3-D Geometry
r
r a b
r a b
a b basis
a b
a b
a b any two non-collinear
a b r a b
r a b
a b cl l
a l b l c
a l b l c
a b c
a b c a b
c b c a
a b c
Example – 4
LOCUSLOCUSLOCUSLOCUSLOCUS 15
Mathematics / Vectors and 3-D Geometry
Solution:
a b c
b c a
c a b
b a b a
a b
a b
a b c
LOCUSLOCUSLOCUSLOCUSLOCUS 16
Mathematics / Vectors and 3-D Geometry
any three non-coplanar
Fig - 20
r
bc
R
O
Q
P
r = OS = OP + OQ + OR
= a + + b c
S
,
a
r
a b c
a b c
a b c
a b c a b c
Linearly independent : na a a avectors
n na a a
n
Linearly dependent na a a a
vectors: n
n na a a
LOCUSLOCUSLOCUSLOCUSLOCUS 17
Mathematics / Vectors and 3-D Geometry
a b c a b c a b c a b c
Solution:
a b c a b c a b c
a b c
a b c
a b c
a b c a b c a b c
Example – 5
