Vector Algebra: Concepts, Components, and Operations, Study notes of Physics

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Chapter 3 Vectors

ƒ

Physics deals with many quantities that have both

ƒ

Size

ƒ

Direction

ƒ

VECTORS !!!!!

ƒ

E.g. Displacement, Velocity, Acceleration, Force,

Torque

x

y

r

ϑ

(

x,y

) (

r,

ϑ

)

x

and

y components

of motion are

independent-”LINEARITY”

j v i v v v v

y

x

y

x

r

θ

θ

cos sin

v

v

v

Componetsv

y x

Unitvectors

Vectors are added or subtracted according

‹ ‹

Ordered pair numbers Ordered pair numbers

( (

x,y

x,y

) ( ) (

r, r,

ϑ ϑ

)

)

Vector Components

y x

y

x

y

y

x

x

a a

a

a

a

a

a

a

=

=

=

θ

θ

θ

tan sin

cos

2

2

2

2

If you use your calculatorto determine angle you will findtan

(-5/7)= -

o

Æ

325

o

VECTOR ALGEBRA

„ „

Scalar multiplication Scalar multiplication

„ „

Vectors are Vectors are

added or subtracted

added or subtracted

according to the according to the

rules for ordered pairs rules for ordered pairs

„ „

( (

a a

xx

,a ,a

y y

) )-

-(

(

b b

xx

,b ,b

y y

) ) coordinates!

coordinates!

a

b

r

r

1.6,

(

)

(

?

1.5)

(4.2,

)

(

−

=

=

=

−

=

=

y

x

y

x

,b

b

b

b

a

,a

a

a

r

r

r

r

See blackboard

Unit VectorNotation

j

ˆ

i ˆ

b

or

2.9)

1.6,

(

b

j

ˆ

ˆi

a

or

1.5)

(4.2,

a

−

=

−

=

−

=

−

=

r

r

r

r

Unit vectors have magnitude 1 andare “unitless” … they only give thedirection!!!!

VECTOR ALGEBRA cont.

„ „

Vectors are Vectors are

added or subtracted

added or subtracted

according to the according to the

rules for ordered pairs rules for ordered pairs

„ „

( (

a a

xx

,a ,a

y y

) )

- -

( (

b b

xx

,b ,b

y y

) )

- -

( (

c c

x x

,c ,c

y y

) )

-- --

coordinates coordinates

! !

or

j

i

c

or

j

i

b

or

j

i

a

v

r

r

Rule for graphical additionis implied!!!!

Displacement Vector: now in 2-D

ƒ

Displacement

ƒ

Three different paths give

the same displacement

Adding more than two vectors graphically

ADDITIONAL VECTOR

PROPERTIES

‹

A vector can be moved (in a diagram) solong as the magnitude and direction isunchanged

‹ ‹

Vectors may be expressed as ordered Vectors may be expressed as ordered

numbers, polar form or in unit vector form numbers, polar form or in unit vector form

‹ ‹

Vector subtraction Vector subtraction

may be accomplished by may be accomplished by

multiplying the subtracted vector by multiplying the subtracted vector by

1 and 1 and

using the technique for adding using the technique for adding

Ant Example

Find resultant Find the vectorhome

Vector Multiplication

„ „

Vectors can be multiplied in two ways

Vectors can be multiplied in two ways

‹ ‹

A dot product of two vectors results in a A dot product of two vectors results in a

scalar scalar

‹ ‹

A cross product of a vector results in A cross product of a vector results in

another another

vector vector

„ „

Vectors are NEVER divided!

Vectors are NEVER divided!

b

a

c

r

r

⋅

=

b

a

c

r

r

r

×

=

Some Properties of the Dot

Product

„ „

Dot products commute

Dot products commute

„ „

The square of a vector

The square of a vector

„ „

Unit vector products

Unit vector products

A

B

B

A

r

r

r

r

⋅

=

⋅

k i k j j i k k j j i i

EXAMPLE 2

What is the dot product of

)

155

,

0

.

2

(

ˆ 0. 3 ˆ 0. 5

o

r r

=

=

B

j

i

A