Chapter 7 Vector algebra
7.1 Scalars and vectors
 Scalar: a quantity specified by its magnitude,
for example: temperature, time, mass, and density
 Vector: a quantity specified by a magnitude and a direction,
for example: force, momentum, and electric field
Vector

a
Chapter 7 Vector algebra
7.2 Addition and subtraction of vectors

a
   
C ommutativ
e: a b  b a
  
  
Associativ
e:
a  (b  c )  (a  b )  c

b
 
ab

  
a  b  a  ( b )
 
ab

a

b

b
 
ba
 
ab

a

b
Chapter 7 Vector algebra
7.3 Multiplication by a scalar



( )a   ( a )   (a )

 

 (a  b )  a  b



(   )a  a  a
x

a

a
Ex: A point P divides a line segment AB in the ratio λ: μ. If the position
vectors of the point A and B are a and b respectively, find the position
vector of the point P.
   
OP  a  AP  a 

a



 
(b  a )

AB

 
 
 
 
 (1 
)a 
b
a
b




B

P

b


p
A

a
O
Chapter 7 Vector algebra

Ex: The vertices of triangle ABC have position vector a, b and c relative
to some origin O. Find the position vector of the centroid G of the triangle.
  1 1
  1 1
OD  d  a  b , OE  e  a  c
2
2
2
2
th eposi ti onve ctorof a ge n e ralpoi n ton th e
l i n eC D th atdi vi de sth e l i n ei n th erati o : (1 -  )
E



 1  
rCD  (1   )c   d  (1   )c   (a  b )
2

e
th epoi n tve ctoron l i n eBE i s
A


1  


G
rBE  (1   )b  e  (1   )b   (a  c )
2


for poi n tG : rCD  rBE

 1  
 1  

c
 (1   )c   (a  b )  (1   )b   (a  c )
a
 D
2
2
d

1
1
2
   ,   1  , 1         
b
2
2
3
 
  
 OG  g  (a  b  c ) / 3
C
F
B
Chapter 7 Vector algebra
7.4
7.4Basis
Basisvectors
vector and
and components
components




a  a1e1  a2e2  a3e3
 

* thre eve ctorse1 , e2 , ande3 forma basis

* a1 , a2 anda3 are thecompone ntsof the ve ctora with
re spe ctto thebasis

A basis set must
(1)
have as many basis vectors as the number of dimension
(2)
be such that the basis vectors are linearly independent



c1e1  c2e2  ......... c N e N  0 exceptc1  c2  ...  c N  0

In Cartesian coordinate system ( x, y, z )

a  a x iˆ  a y ˆj  a z kˆ  (a x , a y , a z )
 
a  b  (a x  bx )iˆ  (a y  b y ) ˆj  (a z  bz )kˆ
Chapter 7 Vector algebra
7.5 Magnitude of a vector


a | a | a x2  a 2y  a z2 is themagnitudeof ve ctora
 
unitve ctoraˆ  a / | a |
7.6 Multiplication of vectors


(1) scalar product a  b

(2) vector product a  b
(1) Scalar product:
 
a  b  abcos 0    


b cos is theproje ctionof b ontothedire ctionof a
 The Cartesisn basis vectors ˆ ˆ
i , j andkˆ
are mutually orthogonal iˆ  iˆ  ˆj  ˆj  kˆ  kˆ  1
iˆ  ˆj  ˆj  kˆ  kˆ  iˆ  0
 
Ex: work: W  F  r
 
potential energy: E   m  B

b
O 
b cos 

a
Chapter 7 Vector algebra
Commutative and distributive:
   
ab  b a
  
   
a  (b  c )  a  b  a  c
In terms of the components, the scalar
product is given by
 
a  b  (a iˆ  a ˆj  a kˆ )  (b iˆ  b ˆj  b kˆ )
x
y
z
x
y
z
 a x bx  a y b y  a z bz

Ex: Find the angle between the vector a  iˆ  2 ˆj  3kˆ

and b  2iˆ  3 ˆj  4kˆ
 
 
ab
cos 
a  b  1  2  2  3  3  4  20
ab
a  12  2 2  3 2  14 b  2 2  3 2  4 2  29
20
cos 
 0.9926   0.12 rad
14 29
Chapter 7 Vector algebra
 direction cosines of vector a

a  iˆ a x
cos x 

a
a

a  ˆj a y
cos y 

a
a

a  kˆ a z
cos z 

a
a
 scalar product for vectors with complex components
 
a  b  a *x bx  a *y b y  a *z bz
 *
 
a  b  (b  a )

 
*
(a )  b   a  b


 
a  (b )  a  b

 
a | a | a  a
Chapter 7 Vector algebra
(1) Vector product:
 
 
a  b  magnitudeis | a || b | sin
 
ab


dire ctionpe rpe ndicu
lar to both a andb

b

Properties:
      
(a  b )  c  a  c  b  c
 
 
b  a  (a  b )
     
(a  b )  c  a  (b  c )
 
aa  0

 

a  b  0  a is paralle or
l an tiparall
e l to b

F
Ex: The moment or torque about O is
  
  
  r  F and|  || r || F | sin

O

r

a
Chapter 7 Vector algebra
Ex: If a solid body rotates about some axis, the velocity of any point
in the body with position vector r is v    r .
For the basis vector in Cartesian coordinate:
iˆ  iˆ  ˆj  ˆj  kˆ  kˆ  0
iˆ  ˆj   ˆj  iˆ  kˆ
ˆj  kˆ   kˆ  ˆj  iˆ
kˆ  iˆ   iˆ  kˆ  ˆj
 
 a  b  (a y bz  a z b y )iˆ  (a z bx  a x bz ) ˆj  (a x b y  a y bx )kˆ
iˆ
ˆj
kˆ
 ax
ay
az
bx
by
bz
Chapter 7 Vector algebra

Ex: a  iˆ  2 ˆj  3kˆ , b  4iˆ  5 ˆj  6kˆ find a  b and the area of
parallelogram.
iˆ
ˆj
kˆ

b
 
a  b  1 2 3  3iˆ  6 ˆj  3kˆ


a
1  
 
 2  | a || b | sin | a  b |
2
4 5 6
 
A | a  b | ( 3) 2  6 2  ( 3) 2  54
 Scalar triple product a  (b  c )  [a, b, c]
  
v  a  b  v  ab si n
 
v  c  vc cos c cos  OP
    
 (a  b )  c  v  c  (ab si n )(c ) cos

| b | sin

v
 vol u m eof a paral l e l eippe d
O
  
 

a  (b  c )  0  a , b andc are coplanar

c



b

a
Chapter 7 Vector algebra
a x a y az c x c y cz
bx b y bz



a  (b  c )  bx b y bz  a x a y a z  c x c y c z
c x c y cz
bx b y bz a x a y a z
  
  
  
  
  
 c  (a  b )  b  (c  a )   a  (c  b )   c  (b  a )   b  (a  c )
Useful formulas:

 
   
   
(1) (a  b )  (c  d )  (a  c )(b  d )  (a  d )(b  c ) Lagran ges' i de n ti ty
  
     
(2) a  (b  c )  (a  c )b  (a  b )c
        
(3) a  (b  c )  b  (c  a )  c  (a  b )  0
Some basic operations:
(1) Kron e ck e rde lta:  ij  1 if i  j
 0 if i  j
(2) Le vi - C ivitasym bol:
εijk  1 e ve npe rm u tatio
n
   ijk mnk   im jn   in jm
k
 1 odd pe rm u tatio
n
 0 an y two of i , j , an d k are e qu al
Chapter 7 Vector algebra
 
a  b   ai b j ij   a i bi
i
j
for 3D i  j  k  1,2,3
i
 
(a  b ) k   a i b j  ijk
i
j
  
     
Ex: Show that a  (b  c )  (a  c )b  (a  b )c
  
Proof: [a  (b  c )]k

  a i (b  c ) j  ijk    a i  bm cn mnj  ijk
i
j
i
j
m
n
  a i bm cn (  mnj  ikj )    a i bm cn [ mi nk   mk  ni ]
i , j m ,n
i
m ,n
   a i bi ck   a i bk ci  (  a i ci )bk  (  a i bi )ck
i
i
     
 [(a  c )b  (a  b )c ]k
i
i
Chapter 7 Vector algebra
7.7 Equations of lines, planes and sphere
Equation of a line:
A line passing through the fixed point A with position vector a and having a

direction b , the position vector r of a general point R on the line is


 
 
 
  
r  a  b  (r  a )  b  (r  a )  b  b  b  0
  
Lineequation: (r  a )  b  0

b
R
A

r

a
O
Chapter 7 Vector algebra
nˆ
Equation of a plane:

A : fi xe dpoi n ton a pl an e ,re pre se n te
d by a ve ctora

R : ge n e ralpoi n ton a pl an e ,re pre se n te
d by a ve ctorr
nˆ : th eu n i tn orm alve ctorof a pl an e
 


( r  a )  nˆ  0  r  nˆ  a  nˆ  d

r  xiˆ  yˆj  zkˆ an d nˆ  liˆ  mˆj  nkˆ
 lx  my  nz  d
R
A

a
d

r
O
pl an ee qu ati on
The equation of a plane containing points A,
B and C with position vectors a, b, andc
 
 
 
r  a   (b  a )   (c  a )
 


 r  a   b  c




( nˆ  r )  ( nˆ  a )  ( nˆ  b )  ( nˆ  c )  d




 nˆ  r   ( nˆ  a )   ( nˆ  b )   ( nˆ  c )
 d   d  d  d        1
 
ba
B
 
ca
A

a

b
O

c
C
Chapter 7 Vector algebra
Ex: Find the direction of the line of intersection of the two planes
x+3y-z=5 and 2x-y+4z=3.


Normal vector of the planes are n1  iˆ  3 ˆj  kˆ n2  2iˆ  ˆj  4kˆ
The direction vector of line is along the direction of
iˆ
ˆj
kˆ
 
n1  n2  1 3  1  10iˆ  6 ˆj  8kˆ
2 1 4
 
r c

r
Equation of a sphere with radius a:
 
   
| r  c |2  (r  c )  (r  c )  a 2

c
O
a
Chapter 7 Vector algebra
Ex: Find the radius  of the circle that is the intersection of the plane nˆ  r  p
and the sphere of radius a centered on the point with position vector c .
 
Th e sph e ree qu ation: | r  c |2  a 2
  2
Th e in te rse cti
n g circlee qu ation: | r  b |   2

c : th epositionve ctorof th ece n e trof a sph e re

nˆ
b : th epositionve ctorof th ece n te rof th ecircle

r : a positionve ctroon th e in te rse cti
n g circle
 
 

plane
(b  c ) || nˆ  b  c  nˆ
b
 
  2  | b  c |2  a 2   2  a 2   2

 
c

2
2

 b  c  nˆ  c  a   nˆ
r
  2    
 
2
| r  b |  ( r  b )  ( r  b )  r  2r  b  b 2
 

 r 2 - 2r  (c  a 2   2 nˆ )  c 2  2(c  nˆ ) a 2   2  a 2   2   2
O
 
 

for | r  c |2  r 2  2r  c  c 2  a 2 an d nˆ  r  p


 p - (c  nˆ )  a 2   2    a 2  ( p  c  nˆ ) 2
Chapter 7 Vector algebra
7.8 Using vectors to find distances
P
 
pa
The minimum distance from a point to a line
 
 
d | p  a | sin | ( p  a )  bˆ |
A

a

p
 
b
O
Ex: Find the minimum distance
from the point P with coordinate


 
(1,2,1) to the line r  a  b , where a  iˆ  ˆj  kˆ , b  2iˆ  ˆj  3kˆ

b
1

bˆ  
( 2iˆ  ˆj  3kˆ ), p  iˆ  2 ˆj  kˆ
b
14
1
1
 
( p  a )  bˆ  ˆj 
[2iˆ  ˆj  3kˆ ] 
[2kˆ  3iˆ ]
14
14
d 
1 2
13
(2  32 ) 
14
14
d
Chapter 7 Vector algebra
P
The minimum distance from a point to a plane
 
d  (a  p)  nˆ
* The signof d de pe ndson whichsideof the
planeP is situate d.
d
nˆ

p
O
 
pa

a
Ex: Find the distance from the point P with coordinate (1,2,3) to the plane that
contains the point A, B and C having coordinates (0,1,0), (2,3,1) and (5,7,2).
 
 
b  a  2iˆ  2 ˆj  kˆ , c  a  5iˆ  6 ˆj  2kˆ
 

 
n  (b  a )  (c  a )  2iˆ  ˆj  2kˆ
 
nˆ  n / | n | ( 2iˆ  ˆj  2kˆ ) / 3

 
d  (a  p)  nˆ  (  iˆ  ˆj  3kˆ )  ( 2iˆ  ˆj  2kˆ ) / 3  5 / 3
for P(0,0,0),d  1 / 3  P(1,2,3)i s on th eoppositesideof th eplan efrom th eorigin
Chapter 7 Vector algebra

b
The minimum distance from a line to
 
a line
ab
 
nˆ     d | ( p  q )  nˆ |
| ab |
Q

q
 
q p
P

p
Ex: A line is inclined at equal angles to the x-, y- and z-axis and pass
through the origin. Another line passes through the points (1,2,4) and
(0,0,1). Find the minimum distance between the two lines.
 

r1  0   ( iˆ  ˆj  kˆ ), r2  kˆ   ( iˆ  2 ˆj  3kˆ )

n  ( iˆ  ˆj  kˆ )  ( iˆ  2 ˆj  3kˆ )  iˆ  2 ˆj  kˆ
nˆ  ( iˆ  2 ˆj  kˆ ) / 6
 
p  q  kˆ  d | kˆ  ( iˆ  2 ˆj  kˆ ) / 6 | 1 / 6
nˆ

a
Chapter 7 Vector algebra
The distance from a line to a plane

 
r  a  b

(1) A lineis notparalle to
l a plane b  nˆ  0, d  0

 
(2) A lineis paralle to
l a plane b  nˆ  0  d | (a  r )  nˆ |

b
nˆ
 
ar

r
O


 
 ˆ
ˆ
ˆ
Ex: A line is given by r  a  b, where a  i  2 j  3kand b  4iˆ  5 ˆj  6kˆ
Find the coordinates of the point P at which the line intersects the plane
x+2y+3z=6.
 

n  iˆ  2 ˆj  3kˆ an d b  n  4  10  18  0

 
r  a  b  xiˆ  yˆj  zk  (4  1)iˆ  ( 2  5 ) ˆj  ( 3  6 )kˆ
x  4  1, y  2  5 an d z  3  6 pu t in toplan ee q.    -1/4
 x  0, y  3 / 4 an d z  3 / 2  (0,3 / 4,3 / 2) is th ein te rse cti
n g poin t

a
Chapter 7 Vector algebra
7.9 Reciprocal vectors
   and  '  '  ' are called reciprocal sets if
The twosets
of
vectors
a
, b, c
a ,b ,c
'  '
 '
aa  b b  c c  1
'  '  '  '  '  ' 
a b  a c  b a  b c  c a  c b  0
 
 
 

bc
c a
ab


 a'    
b'    
c'    
a  (b  c )
a  (b  c )
a  (b  c )
  
 

whe re a  (b  c )  0, a , b andc are not coplanar
 

 '  '  '
if a , b andc are m utuallyorthogonalunitve ctorsthe na  a , b  b , c  c



ˆ
Ex: Construct the reciprocal vector of a  2i , b  ˆj  kˆ andc  iˆ  kˆ
  
a  (b  c )  2iˆ  [( ˆj  kˆ )  ( iˆ  kˆ )]  2

a '  ( ˆj  kˆ )  ( iˆ  kˆ ) / 2  ( iˆ  ˆj  kˆ ) / 2
'
b  ( iˆ  kˆ )  2iˆ / 2  ˆj

c '  2iˆ  ( ˆj  kˆ ) / 2   ˆj  kˆ
Chapter 7 Vector algebra

Define the components of a vector a with respect basis vectors eˆ1 , eˆ2 andeˆ3
that are not mutually orthogonal.
(1) For C arte si anbai sive ctoriˆ , ˆj an d kˆ




a  (a  iˆ )iˆ  (a  ˆj ) ˆj  (a  kˆ )kˆ
 

(2) Foe n on- orth on ormlabasi se1 , e2 an de3 , i tsre ci procal
 

basi sve ctori s e1' , e2' an de3'

a  a1eˆ1  a 2 eˆ 2  a 3 eˆ 3

 
a  e1'  a1eˆ1  eˆ1'  a 2 e2  eˆ1'  a 3 eˆ 3  eˆ1'  a1


a 2  a  eˆ 2'
a 3  a  eˆ 3'




 a  (a  eˆ1' )eˆ1  (a  eˆ 2' )eˆ 2  (a  eˆ 3' )eˆ 3