PH 221-1D Spring 2013

Vectors

Lecture 4,5

Chapter 3 (Halliday/Resnick/Walker, Fundamentals of Physics 9th edition)

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Chapter 3 Vectors

In Physics we have parameters that can be completely described by a number and are known as “scalars” .Temperature, and mass are such parameters.

Other physical parameters require additional information about direction and are known as “vectors” . Examples of vectors are displacement, velocity and acceleration.

In this chapter we learn the basic mathematical language to describe vectors. In particular we will learn the following:

Geometric vector addition and subtraction Resolving a vector into its components The notion of a unit vector Add and subtract vectors by components Multiplication of a vector by a scalar The scalar (dot) product of two vectors The vector (cross) product of two vectors

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An example of a vector is the displacement vector which describes the change in position of an object as it moves from point A to point B. This is represented by an arrow that points from point A to point B. The length of the arrow is proportional to the displacement magnitude. The direction of the arrow indicated the displacement direction.

The three arrows from A to B, from A' to B', and from A'' to B'', have the same magnitude and direction. A vector can be shifted without changing its value if its length and direction are not changed.

In books vectors are written in two ways:

Method 1: (using an arrow above)

Method 2: a (using bold face print)

The magnitude of the vector is indicated by italic print: a

a

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Geometric vector Addition

Sketch vector using an appropriate scale

Sketch vector using the same scale

Place the tail of at the tip of The vector starts from the tail of

and terminates at the tip of

a

b

b as a

s a b

Vector addition is When there are more than two vectors, we cangroup them in any order as we add the

commutative

associative law

Negative of a given vector

m

( ) ( ) ( )

b

a b b a

a b c a b c

b b

has the same magnitude as but opposite direction

b b

Tail‐to‐head method

Parallelogram method

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Geometric vector Subtraction

We write:

From vector

Then we add to vector

We thus reduce vector subtraction tovector addition which we know how to do

we find

d a b a b

b bb

b

a

d a

Note: We can add and subtract vectors using the method of components. For many applications this is a more convenient method

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A B

C

A component of a vector along an axis is the projection of the vector on this axis. For example is the projection of along the x-axis. The component is defined by drawing straight lines fr

x

x

aa a

om the tail and tip of the vector which are perpendicular to the x-axis. From triangle ABC the x- and y-components of vector are given by the

cos , equations:

sin Component Nx ya a aa

a

a

2 2

If we know and we can determine and .

From triangl

, t

otat

an

e ABC

ion

Magnitude-angle Notat

we h

ion

a e

v :x y

yx y

x

a a

a a

a

aa

a

6

7

A vector a can be represented in the magnitude-angle notation (a, ), where 2 2

x ya a a is the magnitude and

1tan y

x

aa

is the angle a makes with the positive x axis. (a) Given Ax = 25.0 m and Ay = 40.0 m, 2 2( 25.0 m) (40.0 m) 47.2 mA (b) Recalling that tan = tan ( + 180°), tan–1 [(40.0 m)/ (– 25.0 m)] = – 58° or 122°. Noting that the vector is in the third quadrant (by the signs of its x and y components) we see that 122° is the correct answer.

Problem 1. The x component of vector is -25.0 m and the y component

is +40.0 m. (a) What is the magnitude of ? (b) What is the angle between

the direction of and the positive direction of x?

A

A

A

Unit VectorsA unit vector is defined as vector that has magnitude equal to 1 and points in a particular direction.

Its sole purpose is to point in a particular direction. The unit vectors along the x, y, and z axes are labe

ˆˆ ˆ

led

, , and , respectively.i j k

Unit vectors are used to express other vectors

For example vectors and can be written as:

and ˆ ˆThe quantitie

ˆ ˆ ˆ ˆ

vector compon

s and are called

the oent f vector s .x y

x y x ya a i a j b b

a b

a i a j

i j

a

b

x y

Thequantities and are scalars, called the

of scalar components componentsor simply .

a a

a 8

x

O

y

a

r b

Adding Vectors by Components

ˆ ˆ ˆ ˆWe are given two vectors and

We want to calculate the vector sum ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )

The components and are given by the equa

x y x y

x y x y x x y y x y

x y

a a i a j b b i b j

r a b a i a j b i b j a b i a b j r i r j

r r

tions:

and x x x y y yr a b r a b

9

10

P rob lem 8 . A car is d riven east for a d istance of 50 km , then north for 30 km ,and then in a d irec tion 30 east of north for 25km . Ske tch the vec tor d iagramand de te rm ine (a ) the m agn itude and (b ) the an

gle of the ca r 's to ta l d isp lacem en t from its sta rting poin t.

r

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All distances in this solution are understood to be in meters. (a) ˆ ˆ ˆ ˆ ˆ ˆ[4.0 ( 1.0)] i [( 3.0) 1.0] j (1.0 4.0)k (3.0i 2.0j 5.0k) m.a b

(b) ˆ ˆ ˆ ˆ ˆ ˆ[4.0 ( 1.0)]i [( 3.0) 1.0]j (1.0 4.0)k (5.0 i 4.0j 3.0k) m.a b

(c) The requirement

a b c 0 leads to c b a , which we note is the opposite of

what we found in part (b). Thus, ˆ ˆ ˆ( 5.0i 4.0 j 3.0k) m.c

Problem 13. Two vectors are given byˆˆ ˆ(4.0 ) (3.0 ) (1.0 )

ˆˆ ˆ( 1.0 ) (1.0 ) (4.0 )

In unit-vector notation, find (a)

(b) , and (c) a third vector such that 0.

a m i m j m k

b m i m j m k

a b

a b c a b c

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As a vector addition problem, we express the situation (described in the problemstatement) as A

+ B

= (3A) j^ , where A

= A i^ and B = 7.0 m. Since i^ j^ we may

use the Pythagorean theorem to express B in terms of the magnitudes of the other twovectors:

B = (3A)2 + A2 A = 110

B = 2.2 m .

Problem 26. Vector , which is directed along an axis, is to be

added to vector , which has a magnitude of 7.0 m. The sum is a third vector that is directed along the axis, with a magnitude that

A x

By

is 3.0 times

that of . What is the magnitude of ? A A

x

O

y

a

d

b Subtracting Vectors by Components

ˆ ˆ ˆ ˆWe are given two vectors and

We want to calculate the vector difference ˆ ˆ

The components and of are given by the equations:

and

x y x y

x y

x y

x x x

a a i a j b b i b j

d a b d i d j

d d

a b

d

d

y y yd a b

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Multiplication of a vector by a scalar r esults in a new vector The magnitude of the new vector is given by

Multiplying a Vecto

:

If 0 vector has the

r by a Scalar

| |b sa s b sa

ab

s b

The S

same

calar

direction

Product o

as vector

If 0

f two

vector has a direction opposite to that of vector

The scalar product of two vectors and is given by:

Vectors

=

a

s b a

a b a

a b

b

The scalar product of two vectors is also known as the product. When two vectors are in unit-vector notation, we write their

cos "dot"

ˆ ˆˆ ˆ ˆ ˆ(

dot

product as

w

(

h

) )x y z x y z

ab

a b a i a j a k b i b j b k

ich we can expand acording to the distributive law: each vector component of the first vector is to be dotted with each vector component of the second vector.By doing so the scalar product of two vectors is given by the equati

on:

= x x y y z za b a b a b a b 14

The vector product of the vectors and is a vector The magnitude of is given

The Vector Pro

by the equation:

The direction of is perpend

duct of two Vectors

sicular

in

c a b a bc

cc

cab

right hand

to the plane P defined

by the vectors and The sense of the vector is given by the :

a. Place the vectors and tail to tailb. Rotate in the plane P along

rule

the shortest an

a bc

a ba

gle

so that it coincides with c. Rotate the fingers of the right hand in the same directiond. The thumb of the right hand gives the sense of The vector product of two vectors is also known as

b

c

"crossthe p" roduct

The cumulative does not applyto a vector product.

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In unit-vector notation, we write

which can be expanded according to the distibutive law,each component of the first vector is to be cross

The Vector Product

ˆ ˆˆ ˆ ˆ ˆ(

ed wi

) )

t

(x y z x y za b a i a j a k b i b j b k

ˆ ˆˆ ˆ ˆ ˆ( ) (

h eac

)ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ

h compon

ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆ

ent of the secon

ˆˆ

d vector

ˆ( ) ( ) ( )ˆ0

.

x y z x y z

x x x y x z y x y y y z

z x z y z z

x y x

a b a i a j a k b i b j b k

a i b i a i b j a i b k a j b i a j b j a j b k

a k b i a k b j a k b k

a b k a

A determinant can also be

ˆˆ ˆ ˆ

used

ˆ0 0ˆˆ ˆ( ) ( ) ( )

ˆˆ ˆˆˆ ˆ

ˆ( ) ( )

z y x y z z x z y

y z y z z x z x x y x y

y z x yx zx y z

y z x yx zx y z

y z y z z x z x

b j a b k a b i a b j a b i

a b b a i a b b a j a b b a k

i j ka a a aa a

a b b a a a a i j kb b b bb b

b b b

a b b a i a b b a

Note: The order of the two vectors in the cross product is import

ˆˆ

an

(

t

)x y x yj a b b a k

b a a b

To check whether any xyz coordinatesystem is a right-handed coordinate system, use the right-hand

rule for the cross product with this system. If your fingets sweep (positive direction of

i j k

i

x) into

(positive direction of y) with the outstretched thump pointing in the positive direction of z, then the system is right-handed.

j

16

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Using the fact that ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆi j k, j k i, k i j or we obtain

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 2 2.00i 3.00j 4.00k 3.00i 4.00j 2.00k 44.0i 16.0j 34.0k.A B

Next, making use of ˆ ˆ ˆ ˆ ˆ ˆi i = j j = k k = 1ˆ ˆ ˆ ˆ ˆ ˆi j = j k = k i = 0 or

we have

ˆ ˆ ˆ ˆ ˆ3 2 3 7.00 i 8.00 j 44.0 i 16.0 j 34.0 k3[(7.00) (44.0)+( 8.00) (16.0) (0) (34.0)] 540.

C A B

P ro b le m 4 0 . F o r th e fo llo w in g th re e v e c to rs , w h a t is

3 ( 2 ) ?ˆˆ ˆ2 .0 0 3 .0 0 4 .0 0

ˆˆ ˆ3 .0 0 4 .0 0 2 .0 0ˆ ˆ7 .0 0 8 .0 0

C A B

A i j k

B i j k

C i j

ˆˆ ˆˆˆ ˆ

ˆˆ ˆ( ) ( ) ( )

y z x yx zx y z

y z x yx zx y z

y z y z z x z x x y x y

i j ka a a aa a

a b a a a i j kb b b bb b

b b b

a b b a i a b b a j a b b a k

= x x y y z za b a b a b a b