Phy 221 2005S Lecture 2 Goals of Lecture 2 Introduce Interactive Learning Segments and try a few The Language of Vectors: –Understand conventions used

Phy 221 2005S Lecture 2

Goals of Lecture 2Goals of Lecture 2

• Introduce Interactive Learning Segments and try a few

• The Language of Vectors: – Understand conventions used in denoting vectors.– Visualize vector operations both from the geometric

and algebraic point of view.

• Dot Products: – Understand geometric and algebraic formulation of the

vector dot product.– Understand some properties of the dot product


What is a VectorWhat is a Vector

• A scalar quantity is one that is represented by a single number (e.g. Mass Length time temperature volume…)

• A vector is a quantity which has both magnitude and direction (e.g. displacement, velocity, force)

• Magnitude: How long is the vector

• Direction: angle counterclockwise from x-axis or other sensible description.

• Geometrically: we represent a vector as an arrow

BasicsBasics

• Equal Vectors: vectors are equal if they have the same magnitude and direction regardless of where the vector “starts”

• Opposite Vectors: Vector are opposite if the magnitude is the same but direction is opposite

• Unit Vectors: is in the direction of but of length 1.

A

BBA

A

BBA

A A


Vector AdditionVector Addition

Geometrically: Parallel transport the tail of B to the head of A. The sum goes from the tail of A to the head of B.

A+B=C

A (5,1)

B(-2,4)C(3,5)

Algebraically: Add the components

A 5 1 B -2 4 C 3 5

Vector addition is commutative and associative


interACTive learning segments interACTive learning segments (ACTs)(ACTs). .

• Get your clickers ready• I will post a question, you should talk to

your neighbor and “vote” on the correct answer with your clicker

• You can change your answer if you change your mind.

• When you click, a box with your clicker number will appear

• At the end of the time, a bar graph will show the summary of results.


ACT: Vector additionACT: Vector addition

• All the vectors below have the same magnitude. Which of the following arrangements will produce the largest resultant when the two vectors are added?

1. 2. 3.


The language of vectorsThe language of vectors

• To keep straight which variables are scalar quantities, it is conventional to draw a little arrow over vector variables. A B C

• If A is a vector, we use A (without arrow) to denote the magnitude.

• Sometimes boldface is used for vector and normal font for magnitude. Thus vector=A; magnitude=A

• More formally, we can indicate magnitude of a vector by vertical bars.

• Thus A=|A|.


Vector AdditionVector Addition

• Geometric:

A

BCA+B=C

• Algebraic: Ax+Bx=Cx

Ay+By=Cy

Az+Bz=Cz

• Subtraction: A-B=D

A

BD

-BNote two ways to think:D is A plus –BD goes from tip of B to tip of A when A and B are rooted at a common point


ComponentsComponents

• The components of a vector can be thought of as the projections along the coordinate axes. These are sometimes called the Cartesian coordinates:

• We can denote A in terms of its components as: A=(Ax,Ay,Az)

x

y

A

Ax

Ay


Unit VectorsUnit Vectors

• A unit vector is a vector of length 1. To indicate a vector is a unit vector, we put a hat on it:

• Some special unit vectors

– The unit vector that points along the x axis is denoted i– The unit vector that points along the y axis is denoted j– The unit vector that points along the z axis is denoted k

^

^

^

i

j

k

^

^

^x

y

z

Any vector can be written in terms of these basic unit vectorsIf A=(Ax,Ay,Az) then

kAjAiAA zyxˆˆˆ

A

Adenotes a unit vector parallel to A


How do I remember which is sin and which is cos?

Polar NotationPolar Notation

• In 2 dimensions, one can also describe a vector by its magnitude and direction.

• The direction is the angle taken counterclockwise from the x axis

A|A|

sin||

cos||

AA

AA

y

x

A

A

A

A

AAA

x

x

y

yx

arccos

arctan

|| 22


Dot ProductsDot Products

• Dot Products are the workhorse of vector analysis

DefinitionAlgebraic

•In terms of components:

zzyyxx BABABABA

Geometric

•If is the angle between A and B:

cos|||| BABA

A

B

Where doesthis come from?


Properties of The Dot ProductProperties of The Dot Product

• The dot product takes two vectors as inputs and produces a scalar as output.

• The dot product is commutative: A•B=B•A• Dot product distributes over vector addition:• The dot product between a vector, A, and a unit, u, vector

gives the projection of A along u.• In particular, the components

of A are the dot products withi, j and k

• The length of a vector can be expressed in terms of the dot product: |A|²=A•A

CABACBA

)(

u

A

A•u

kAAjAAiAA zyxˆ


Some Special CasesSome Special Cases

• Vectors that are going in the same direction

• Vectors that are going in opposite directions

• Perpendicular vectors

ABBA

ABBA

0 BA

Remember

vector #1

vector #2

Dot Product

Scalar

Note: Later in the course we will learn about another kind of product called the “cross product” which takes in two vectors and spits out another vector. Do not get them confused.


Angle Between VectorsAngle Between Vectors

• We can use the geometric definition of the dot product to determine the angle between two vectors:

• This tells the angle between A and B but not the direction of the angle

cos|||| BABA

||||

cosBA

BA

A

B


Example of Vector Algebra:Example of Vector Algebra:Law of cosinesLaw of cosines

Consider a triangle, The sum of the vectors representing the sides is 0

A

BC

C=-(A+B) C²=(A+B)²=(A+B)·(A+B) =A²+B²+2A·B =A²+B²+2AB cos() =A²+B²-2AB cos()

Binomial expansion of dot product: a usefultrick-learn it!

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Phy 221 2005S Lecture 2 Goals of Lecture 2 Introduce Interactive Learning Segments and try a few The Language of Vectors: –Understand conventions used