Transcript mp_ch7
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
ab
a b a ( b )
ab
a
b
b
ba
ab
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:
ab 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
ab
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
ab
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 )
aa 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
ba
B
ca
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
pa
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
pa
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
ab
nˆ d | ( p q ) nˆ |
| ab |
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
ar
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 twosets
of
vectors
a
, b, c
a ,b ,c
' '
'
aa b b c c 1
' ' ' ' ' '
a b a c b a b c c a c b 0
bc
c a
ab
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