Physics 201, Lecture 4 Today’s Topics

n  Vectors (chap 3) n  Scalars and Vectors n  Vector Addition Rule n  Vector in a Coordinator System n  Decomposition of a Vector

n  Expected from preview: n  Scalars and Vectors, Vector Addition, Vector Components,...

About Exam 1 q  When and where

§  Monday Oct 2nd, 5:30-7:00 pm §  125 Ag Hall (all sections).

q  Format §  Closed book §  One 8”x11” formula sheet allowed, must be self prepared. (Absolutely no photocopying/printing of sample problems,

examples, class lectures, HW etc.) §  20 multiple choice questions. §  Bring a calculator (but no computer). Only basic calculation

functionality can be used (see my earlier email for details) §  A B2 pencil is required to do with Scantron

q  Special needs/ conflicts: §  Should be settled by now (except for emergency). §  Only medical/academic excuses are considered. §  All alternative test sessions are in our lab room, only for

approved requests.

Chapters Covered q  Chapter 1: Physics and Measurement. q  Chapter 2: Motion in 1-D

§  Sections 2.1-2.7 .

q  Chapter 3: Vectors (will largely be tested indirectly via physics problems)

q  Chapter 4: Motion in 2-D §  All Sections

An in-lecture review is scheduled on Thursday Sep 28th.

Vectors and Scalars q  A scalar is a quantity with magnitude but no direction

§  Scalar examples: mass, length, temperature, ....␣ q  A vector is a quantity defined by both a magnitude and a direction

§  Vector examples: displacement, velocity, acceleration, force, ...

q  Vectors notations: §  bold face letter: v, x, A, ... §  letter with an “arrow” hat: §  graphically: an arrow with length and direction

§  Magnitude (length) of vector A : |A| (note: |A| is just a number, i.e. a scalar)

,...,, Axv

Equality Vectors and Negative Vector q  Vector equality: same magnitude and same direction

= A = B

q  Negative Vector (-): same magnitude but opposite direction

e.g. These are all identical vectors

A B = - A

negative

Vector Addition Rule q  “Tail to head” rule:

q  Vector addition is commutative: A + B = B + A

Associative Rule of Vector Addition q  When three or more vectors are added together, the result is

independent of how the vectors are grouped. (Associative Law)

A + (B +C ) = (A + B) + C

Vector Subtraction q  Vector subtraction is a special case of vector addition A – B is equivalent to A + (-B)

q  Graphically:

q  Tip: A–B is a vector drawing from the head of B to the head of A

+ -B

A

- A

B

Quick Quiz: Relative Velocity q  Consider a situation in which a passenger observes rain drops

inside a running bus Let vrE be velocity of rain w.r.t Earth vbE be velocity of bus w.r.t Earth vrb be velocity of rain w.r.t. bus Physics suggests an graphical relationship as shown in graph above. (later in lecture 6) Which of the followings represents the matching formula?

vbE

vrb vrE

vrb =vrE +vbE

vrb =vrE −vbE

vrb =vbE −

vrE

A B C

Displacement in Vector Form q  1-D form

§  Δx = xf-xi §  direction sign (+/-)

q  Full vector form (1,2,3-D)

§  Δr usually not in line with ri or rf

if rrr −=Δ

x xi=3 m xf=1 m

Δx = xf- xi = -2m

Thursday: Kinematics in 2/3 D

Vector Scaling q  When a vector is multiplied ( or divided) by a positive number,

its magnitude is multiplied (or divided) by that number while its direction remains unchanged.

q  When a vector is multiplied ( or divided) by a negative number, its magnitude is multiplied (or divided) by (the absolute value of) that number and its direction is reversed.

q  Examples:

x 2 = and x (-0.5) =

A 2A A -0.5A

Unit Vector q  Quiz: Let vector B = A/|A|, what is the length of B? (hint: do you remember what |A| stands for?) Choices: 1 , undefined q  Unit vector is a (dimensionless) vector with length =1

§  Notation of unit vector

§  Little Exercise: verify

§  Graphically: A =

A

A =A/|A| or

A = |A|A

|A| A

Vector and Coordinator Systems q  As we have been doing, vectors are defined independent of any

coordinator system. q  However, it is often helpful to put vectors into coordinator

systems.

q  Math review: Cartesian and polar coordinator systems:

Cartesian: (x,y) or (x,y,z) Polar: (r, θ ) or (r, θ, φ)

Vector in Cartesian System q  Graphically one can easily see that A = Ax + Ay

q  Take notation:

j Aand iA yxˆˆyx AA ==

j i AAA yxˆˆyx AA +=+=

k j i A AA A:3D in zyxˆˆˆzyx AAA ++=++= (Vector Decomposition)

A Little Trigonometry (2D) q  Converting from length and direction (i.e. angle) to (x,y) :

q  Converting from (x,y) to length and direction:

q  note: 3D is slightly more complicated, but still doable.

θ

θ

sincosAAAA

y

x

=

=

)(tan 1

22

x

y

yx

AA

AAA

−=

+=

θAx = Acosϕ sinθAy = Asinϕ sinθAz = Acosθ

Vector Addition in (x,y,z) system q  Step 1: Write the vectors into x,y,z forms: (de-composition)

q  Step 2: Add the vectors by their components

q  Step 3: Calculate the magnitude and the direction of the

resulting vector if necessary.

k j iB B BB

k j i A AAA

zyx

zyx

ˆˆˆ

ˆˆˆ

zyx

zyx

BBB

AAA

++=++=

++=++=

k)( j)( i)

k j ik j iBARˆˆˆ(

ˆˆˆˆˆˆ

zzyyxx

zyxzyx

BABABA

BBBAAA

+++++=

+++++=+=

Graphically....

and don’t Forget

zzzyyyxxx

zzyyxx

zyxzyx

BARBARBARBABABA

BBBAAA

+=+=+=

+++++=

+++++=+=

k)( j)( i)

k j ik j iBAR

,,

ˆˆˆ(

ˆˆˆˆˆˆ

.),(tan

||

1

222

etcRR

RRRR

x

yxy

zyx

−=

++==

θ

R

Vector Multiplication q  There are two forms of vector multiplication

§  Dot product: A•B ≡ ABcosθ •  Graphically: •  In rectangular coordinates

A•B = AxBx + AyBy + AzBz

§  Cross product: C=AXB = ABsinθ , in a direction defined by right hand rule

Graphically:

•  In rectangular coordinates: C=AxB !à Cx = AyBz-AzBy, Cy=AzBx-AxBz, Cz=AxBy-AyBx

Exercise: Taking a Hike q  A hiker begins a trip by first walking 25.0 km southeast from her car

to reach a tent. Then she walks 40.0 km in a direction 60.0o north of east to reach a tower. §  What is her total displacement?

q  Solution: We need to do vector sum R = A+ B Method 1: do it graphically. Method 2: do it in (x,y) system Follow steps in last slide. Step 1: De-composition Ax = Acos(-45o) = 25 * 0.707 = 17.7 km, Ay = A sin(-45o) = 25 * (-0.707) = -17.7 km Bx = Bcos(60o)= 40 * 0.50 = 20.0 km, By = B sin(60o)= 40 * (0.866) = 34.6 km Step 2: add components separately: Rx= Ax+ Bx= 17.7 + 20.0 = 37.7 km , Ry= Ay+By= -17.7 +34.6 = 17.0 km

km )j i(37.7R ˆ0.17ˆ += After class: Find the length and angle of R

Math Background: Product of Two Vectors q  Recall: Vector is a quantity with a magnitude and a direction

q  A (or ): = (Ax, Ay, Az)

q  Recall: Addition/Subtraction can be defined between two vectors: §  A + B = (Ax+ Bx, Ay + By, Az + Bz) §  A - B = (Ax - Bx, Ay - By, Az - Bz)

q  There are two fashions of multiplication between two vectors §  Scalar product (dot product)

§  Vector product (cross product)

(next slide)

A

A•B = (AxBx + AyBy + AzBz ) = ABcosθ

A×B = ?