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Vector Product Results in a vector
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Vector Product. Results in a vector. Dot product (Scalar product). Results in a scalar a · b = a x b x +a y b y +a z b z. Scalar. Vector Product. Results in a vector. =. Properties…. a x b = - b x a a x a = 0 a x b = 0 if a and b are parallel. - PowerPoint PPT Presentation

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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

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