Physics 201 2 Vectors Coordinate systems Vectors 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.

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

Measurement of Angles s r in Radians q 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.

Properties of vectors r denoted by : v or magnitude denoted by = v length : v or can always represent a vector by a directed line segment: y x v

• Two vectors are equal if they have • same length • same direction = parallel transport is moving vector without changing length or direction

Addition V 2 tip V 1 tail +

V 1 V 2 V 1 + V 2

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

special unit vectors k i j

components of vectors V yj xi V = xi + yj

V yj xi components of vectors

v q x i yj

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

• 1 -1 correspondence between vectors their coordinates • V = xi + yj =(x, y) and Addition : ( ) b =b i +b j º(b , b ) a +b ={a +b }i+{a +b }jº(a +b , a +b ) a = axi +ayj º ax , ay x x x y y

b a Scalar Product

coordinate form of scalar product

V j V = xi + yj i V· i = x = V Cos (q ) V· j = y = V Sin (q )

Polar form of vectors v = v x i + v y j = v cos q i + v sin q j = v (cos q i + sin q j) º v (cos q, sin q ) now (cos q i + sin q j) = cos 2 q + sin 2 q = 1 Thus vˆ = cos q i + sin q j is a unit vector in the direction of v and v = v vˆ POLAR FORM of the vector v vˆ = v v

Special Vectors (x, y) r ri d rf

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