Post on 09-Jan-2016
description
Vector Product
Results in a vector
Dot product (Scalar product)
Results in a scalar
a · b = axbx+ayby+azbz
Scalar
Vector Product
Results in a vector
zyx
zyx
bbb
aaa
kji
ba
ˆˆˆ
kji ˆˆˆyx
yx
zx
zx
zy
zy
bb
aa
bb
aa
bb
aa=
Properties…
a x b = - b x a a x a = 0 a x b = 0 if a and b are parallel. a x (b + c) = a x b + a x c a x (λb)= λ a x b
Examples…
i x j = k, j x k = i, k x i = j j x i = - k, i x k = -j, k x j = i i x i = 0
Vector product
c is perpendicular to a and b, in the direction according to the right-handed rule.
c = a x b
a
b
c
θ
Vector product – Direction: right-hand rule
a
b
cc = a x b
θ
b
a
c
Vector product – right-hand rule
a
b
cc = a x b
θ
b
a
c
Vector product – right-hand rule
c = a x b
θb
a
c
a
b
c
Vector Product-magnitude
21
21
1ˆ
0
00
ˆˆˆ
ba
bb
a k
kji
bac
sinˆ bak
c
a
b
θx
y
a = (a1, 0, 0)
b = (b1, b2, 0)
b2
sinbaba
Invariance of axb
The direction of axb is decided according the right-hand rule.
The magnitude of axb is decided by the magnitudes of a and b and the angle between a and b.
a x b is invariant with respect to changes from one right-handed set of axes to another.
sinbaba
Application—Moment (torque) of a force
O
F
R
M
Moment of a force about a point
O
F
R
M = | F | |R |sinθ
dM = R x F
θ
M = | F |d
Component of a vector a in an arbitrary direction s
i
ai ˆxa
asˆsaa
s
as
xax
ssss
ss zyx sss
,,ˆ
--- Unit vector in the direction of s
Example--Component of a Force F in an arbitrary direction s
ssss
ss zyx sss
,,ˆ
kji()kj7
iFs 715)ˆˆˆ3ˆ7
6ˆ3ˆ7
2(ˆ /Fs
Fs
Fs--- Unit vector in the direction of s
kji3F ˆˆˆ k6j3i2s ˆˆˆ
kj7
ikjiss/s ˆ7
6ˆ3ˆ7
27/)ˆ6ˆ3ˆ2(ˆ
FsˆsF
Example--Component of a Moment M in an arbitrary direction s
ssss
ss zyx sss
,,ˆ
MsˆsM
FRM
Ms
Fs --- Unit vector in the direction of s
FRsMs ˆˆsM---- Scalar Triple product
Scalar Triple Productc)(ba
kji ˆˆˆyx
yx
zx
zx
zy
zy
cc
bb
bc
bb
cc
bb
Scalar
zyx
zyx
ccc
bbb
kji
cb
ˆˆˆ
c)(ba
kjikji ˆˆˆ)ˆˆˆ(
yx
yx
zx
zx
zy
zyzyx cc
bb
bc
bb
cc
bbaaa
zyx
yxy
zx
zxx
zy
zya
cc
bba
bc
bba
cc
bb
zyx
zyx
ccc
bbbzyx aaa
Volume of a parallelepiped
coscbac)(ba
sincos cba
θ
α
a
b
c
= Volume of the parallelepiped.
A
B
C
D
E
F
G
Hbxc
θ
Moment of a force about an axis
ssss
ss zyx sss
,,ˆ
FRM F
s
--- Unit vector in the direction of s
FRsMs ˆˆsM---- Moment of F about axis AA’
A’
A
Vector Triple Productc)(ba Vector