Physics 201 2: Vectors Coordinate systems Vectors and scalars Rules of combination for vectors Unit vectors Components and coordinates Displacement and.

Physics 201 2: Vectors

•Coordinate systems •Vectors and scalars•Rules of combination for vectors•Unit vectors•Components and coordinates•Displacement and position vectors•Differentiating vectors•Kinetic equations of motion in vector form•Scalar (=dot) product of vectors

Coordinate Systems

•1. Fix a reference point : •ORIGIN

•2. Define a set of directed lines that intersect at origin:

•COORDINATE AXES•3. Instructions on how to label point with respect origin and axes.

r

x

yb

a

p

•rectangular cartesian coordinates of point “p” = (a,b)•plane polar coordinates of point “p” = (r,)

Measurement of Angles

r

sin Radians

is measured counterclockwise from

+ x-axis

r

s

Transformation from polar coordinates

to cartesian coordinates

x rcosy rsin

Transformation from cartesian coordinates

to polar coordinates

r x2 y2

tan 1 y

x

;

y

x 0 then

y 0 0,90 y 0 180,270

y

x 0 then

y 0 90,180 y 0 270,360

Vectors and scalars

•Scalar: •has magnitude but no direction

•e.g. mass, temperature, time intervals, .....

•Vector: •has magnitude and direction

•e.g. velocity, force, displacement, ......

•Displacement vector•line segment between final position and initial position.

can always represent a vector by a directed line segment:

x

y

Properties of vectors

denoted by : v or v or v

magnitude = length

denoted by : v or v or v

•Two vectors are equal if they have•same length•same direction

=

parallel transport is

moving vector without changing length or direction

+V1

V2

Addition

tip

tail

V1 V2

V1 + V2

Addition is

Commutative: a b b a

Associative: a (b c) (a b) c

a = vector that has same length as a

but opposite direction

Multiplication by scalar:

ma

m 0 vector in same direction as a

but m times as long

m 0 vector in opposite direction as a

but m times as long

Unit vectors

Any vector that has magnitude 1

i.e. a =1

is a unit vector

special unit vectors

k

ji

V

V = xi + yj

components of vectors

ix

jy

V

components of vectors

jy

ix

x i

y jv

x v cos

y v sin

tan 1 y

x

22 yxv v

coordinates of vectors

V

(x,y)

xi

yj

V=xi + yj

V

•1-1 correspondence between vectors and their coordinates•V = x i + y j =(x, y)

Addition:

aaxiay jax,ay bbxiby jbx,by

abaxb

x i ay by ja

xb

x,ayb

y

a b a bCos

b

a

Scalar Product

a b axbx

+ayby+a

zbz

a a ax2 ay

2 az2 a

2

i j i k j k 0

coordinate form of scalar product

i

j

V

Vxi yj

V i x V Cos

V j y V Sin

Polar form of vectors

v vxi v

yj v cos i v sin j

v cos i sin j v cos , sin

now cos i sin j cos 2 sin 2 1

Thus ˆ v cos i sin j is a unit vector

in the direction of v and

v v ˆ v POLAR FORM of the vector v

ˆ v =vv

Special Vectors

(x,y)r

POSITION VECTOR : r xi yj x, y ri

rf

d

DISPLACEMENT VECTOR: d r f r i

differentiating vectors

differentiate coordinate functions

r t x t i +y t jdr t dt

dx t dt

i dy t dt

j v t

d2r t dt2

d 2x t dt2

i d2y t dt 2

j dv t dt

a t

v t vx t i vy t j

a t ax t iay t j

Vector Kinetic Equations of Motion

r t 1

2at 2 v 0 t r 0

d t 12

at 2 v 0 t

v t at v 0 Kinetic Equations for each component/

coordinate

Solving Problems Involving Vectors

1. Graphically

! Draw all vectors in pencil ! Arrange them tip to tail

! Draw a vector from the tail of the first vector to the tip of the last one.

! measure the angle the vector makes with the positive x-axis

! measure the length of the vector.

! measure the length of its X component

! measure the length of its Y component

2. Algebraically

! write all vectors in terms of their X and Y components

! The X component of the sum of the vectors is the sum of the X components

! The Y component of the sum of the vectors is the sum of the Y components