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Page 1: Vector Product

Vector Product

Results in a vector

Page 2: Vector Product

Dot product (Scalar product)

Results in a scalar

a · b = axbx+ayby+azbz

Scalar

Page 3: Vector Product

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=

Page 4: Vector Product

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

Page 5: Vector Product

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

Page 6: Vector Product

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

Page 7: Vector Product

Vector product – Direction: right-hand rule

a

b

cc = a x b

θ

b

a

c

Page 8: Vector Product

Vector product – right-hand rule

a

b

cc = a x b

θ

b

a

c

Page 9: Vector Product

Vector product – right-hand rule

c = a x b

θb

a

c

a

b

c

Page 10: Vector Product

Vector Product-magnitude

21

21

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

Page 11: Vector Product

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

Page 12: Vector Product

Application—Moment (torque) of a force

O

F

R

M

Page 13: Vector Product

Moment of a force about a point

O

F

R

M = | F | |R |sinθ

dM = R x F

θ

M = | F |d

Page 14: Vector Product

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

Page 15: Vector Product

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

Page 16: Vector Product

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

Page 17: Vector 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

Page 18: Vector Product

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

Page 19: Vector Product

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

Page 20: Vector Product

Vector Triple Productc)(ba Vector