lecture1 vectors

WELCOME TO PHYSICS 101Introduction to Physics 1 !

• Learning to think and reason about the physical world

- Conceptual Understanding- Make connections with real-world applications

• Build a “physical intuition”

• Problem-Solving skills

• This course is not about memorization or mindless use of mathematicswe will use math to think about relationships between quantities.

• This is a course that is very different from high school physics.

Class Themes

What is Physics all about ?• Provides a quantitative understanding of physical

phenomena in our universe

• Based on experimental observations and mathematical analysis

• Used to develop theories that explain the phenomena being studied and relate to other established theories

• Experiments refine our theories about our universe

To be quantitative….• Need standards of measurement for different physical

quantities

• Use SI System for measuringa. length

1 meter = distance travelled by light in vacuum in a certain fraction of second.

b. mass1 kilogram = mass of a cylinder kept in

Int’l Bureau of Weights/Standardsc. time

1 second = a certain # times the period of radiation oscillation of a cesium atom

• Scalar - a physical quantity that is specified by a positive or negative number with a unit.Ex. Temperature, Volume, Mass, Time, Energy

Rules of ordinary arithmetic are used to manipulate scalar quantities

• Vector - is a physical quantity that must be described by a magnitude (number) and unit, plus a direction.

Scalars vs. Vectors

Example of a Vector• A particle travels from A

to B along the path shown by the dotted red line– This is the distance

traveled and is a scalar

• The displacement is the solid line from A to B– The displacement is

independent of the path taken between the two points

– Displacement is a vector

Other vectors: velocity, acceleration, force, momentum

Vectors: Notation and Properties• When handwritten, use an arrow: • When printed • Magnitude | |

Ar

Ar

Ar

Equality of Two Vectors

• Two vectors are equal if they have the same magnitude and the same direction

• All of the vectors shown are equal

Adding Vectors• When adding vectors, their directions must be taken into

account• Units must be the same

(can’t add velocity and acceleration vectors)

Two Methods: 1. “Graphical” Method

- use scale drawings2. “Component or Analytical” Method

- more convenient

Adding Vectors GraphicallyEstablish a scale and coordinate systemDraw vectors to be added “tip to tail”.Measure the resultant R and itsdirection, relative to A or B

θ

Or Use Trigonometry (Sine and Cosine Laws)

Adding Multiple Vectors

“Tip-to-tail” Method is Repeated

Note: Vector Addition is Commutative

A + B = B + A

(Ordering doesn’t matter)

Subtracting Vectors• Negative of a vector

– reverses the vector’s direction

• Continue with standard vector addition procedure

Multiplying/Dividing a Vector by a Scalar

• The result is a vector, “scaled” by the scalar factor• Negative scalar multiplication reverses direction.

Dot/Scalar and Vector Cross Products – very important vector multiplcation will be studied in future chapters

A

2A

- A

Components of a Vector

Rectangular components– projections of the vector

along the x- and y-axes

A vector A can be represented as the sum of its components Ax and Ay .

A = Ax + Ay

θ= cosAxA θ= sinAyA

x

y12y

2x A

AtanandAAA −=θ+=

One can further simplify this with the use of “unit vectors”.

Unit Vectors• A unit vector is a dimensionless

vector with a magnitude of exactly 1.• Unit vectors are used to specify

a direction

kand,j,i

• provide the basis for representing any vectorin the space.

Unit Vectors – why use them ?• The symbols

represent unit vectors in the x, y and z directions

• They form a set of mutually perpendicular vectors

kand,j,i

Unit Vectors in Vector Notation

• is the same as Ax

is the same as Ay etc.

• The complete vector can be expressed as

i

j

kjiA ˆAˆAˆA zyx ++=r

xAr

yAr

Adding Vectors Using Unit Vectors

• Since• Then

• Then Rx = Ax + Bx and Ry = Ay + By

( ) ( )( ) ( )jiR

jijiRˆBAˆBA

ˆBˆBˆAˆA

yyxx

yxyx

+++=

+++=r

r

x

y12y

2x R

RtanRRR −=θ+=

BARrrr

+=

Adding Vectors with Unit Vectors

Adding Vectors Using Unit Vectors – Three Directions

• Using

• Rx = Ax + Bx , Ry = Ay + By and Rz = Az + Bz

etc.RRtanRRRR x1

x2z

2y

2x

−=θ++=

BARrrr

+=

Lecture Problem 1-8

Suppose your hair grows at the rate 1/32 inch per day. Find the rate at whichit grows in nanometers (nm) per second. Because the distance between atoms in a molecule is on the order of 0.1nm, your answer suggests how rapidly layers of atoms are assembled in this protein synthesis.

50x400xHair

400x

Lecture Problem 1-8

Suppose your hair grows at the rate 1/32 inch per day. Find the rate at whichit grows in nanometers (nm) per second. Because the distance between atoms in a molecule is on the order of 0.1nm, your answer suggests how rapidly layers of atoms are assembled in this protein synthesis.

(1/32) inch/day x .0254 m/inch x 109 nm/m x 1 day/24 hr x 1 hr/60 min x 1 min/60 sec

9.2 nm/s ~ 90 layers of atoms/second

50x400xHair

400x

Lecture Problem 1-45Consider the two vectors: A = 3i – 2j and B = -i – 4j. Calculate

a. A + B c. |A + B| e. the directions of A + B, A - Bb. A – B d. |A – B|

Lecture Problem 1-45Consider the two vectors: A = 3i – 2j and B = -i – 4j. Calculate

a. A + B c. |A + B| e. the directions of A + B, A - Bb. A – B d. |A – B|

Solution:a. A + B = 3i – 2j + -i – 4j = 2i – 6j

b A – B = 3i – 2j – ( -i – 4j) = 4i + 2j

c. |A + B| = √(22 + 62) = 6.32

d. |A – B| = √(42 + 22) = 4.47

e. direction of A + B: tan θ = -6/2 θ = -71.6° = 288°

direction of A – B: tan θ = 2/4 θ = 26.6°

Question: What to do if there are 3 vectors to add or subtract ?