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

Measurement of Angles r s in Radians  is measured counterclockwise from + x-axis

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 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

+ V1V1 V2V2 Addition tip tail

V1V1 V2V2 V2V2 V1V1 + V 2

Unit vectors Any vector that has magnitude 1 i.e. a =1 is a unit vector

special unit vectors k j i

V V = x i + y j components of vectors i x j y

V j y i x

x i y j v 

coordinates of vectors V (x,y) xixi yjyj V=xi + yj V

1-1 correspondence between vectors and their coordinates V = x i + y j =(x, y) Addition: a  a x i  a y j  a x,a y  b  b x i  b y j  b x,b y  a  b  a x  b x  i  a y  b y  j  a x  b x,a y  b y 

b a Scalar Product

coordinate form of scalar product

i j V V  xi  yj V  i  x  V Cos    V  j  y  V Sin   

Polar form of vectors v  v x i  v y j  vcos  i  vsin  j  vcos  i  sin  j   vcos ,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 = v v

Special Vectors (x,y) r riri rfrf d

differentiating vectors

Vector Kinetic Equations of Motion

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