introduction to vectors - umass...

Department of Physics and Applied Physics 95.141, Fall 2013, Lecture 3

Course website:

http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Lecture Capture:

http://echo360.uml.edu/danylov2013/physics1fall.html

Lecture 4

Introduction to

vectors

http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

http://echo360.uml.edu/danylov2013/physics1fall.html


Chapter 3. Section 3.1 – 3.5

• Vectors and Scalars

• Addition of Vectors

• Subtraction of Vectors

• Multiplication of a Vector by a Scalar

• Adding Vectors by Components

• Unit Vectors

Outline


Vector and Scalars

e.g. distance, speed, temperature, mass, time, density, volume

r,v ,a

e.g. displacement, velocity, acceleration, force, momentum

has only magnitude

(no need in direction)

Vector quantity Scalar quantity

r

has both direction and magnitude


Addition of Vectors (1D)

If the vectors are in

opposite directions

If the vectors are in the

same direction

For vectors in one dimension, simple addition and subtraction are all that is needed. Easy!!!!


Addition of Vectors (2D). Graphical Methods

Triangle method.

• “Tail-to-Tip” method

• Draw first vector

• Draw second vector, placing tail at tip of first vector

• Arrow from tail of 1st vector to tip of 2nd vector: resultant

𝑨 𝑩

If the motion is in two dimensions, the situation is somewhat

more complicated.


Addition of Vectors (2D). Graphical Methods

Parallelogram method.

• “Parallelogram” method

• The two vectors, 𝐴 𝑎𝑛𝑑 𝐵, are drawn as the sides of the

parallelogram and the resultant, 𝐶 = 𝐴 + 𝐵, is its

diagonal

𝑨

𝑩

Commutative property of vectors 𝑪 = 𝑨 + 𝑩 = 𝑩 + 𝑨


Subtraction of vectors

is a vector with the same magnitude as but in the

opposite direction. So we can rewrite subtraction as addition

B

BA

= A

AB

B

B

𝑪 = 𝑨 − 𝑩

B

So, we add the negative vector.

)( BA

ConcepTest 1 Vector Addition

You are adding vectors of length

20 and 40 units. What is the only

possible resultant magnitude that

you can obtain out of the

following choices?

A) 0

B) 18

C) 37

D) 64

E) 100

ConcepTest 1 Vector Addition

You are adding vectors of length

20 and 40 units. What is the only

possible resultant magnitude that

you can obtain out of the

following choices?

A) 0

B) 18

C) 37

D) 64

E) 100

The minimum resultant occurs when the vectors

are opposite, giving 20 units. The maximum resultant

occurs when the vectors are aligned, giving 60 units.

Anything in between is also possible for angles

between 0° and 180°.

40 20

Min=40-20=20 Max=40+20=60

Resultant is between 20 and 60


Multiplication of a Vector by a Scalar

𝐴

𝐵 = 1.5 𝐴 𝐶 = −2.0 𝐴

A vector can be multiplied by a scalar b(positive); the result is a

vector that has the same direction but a magnitude .

𝐴 𝐵 𝑏𝐴

If b is negative, the resultant vector points in the opposite direction.


Addition of three or more vectors

Can use “tip to tail” for more than 2 vectors

+ + = 𝑨

𝑩

𝑪 𝑨 𝑩

𝑪

𝑫 = 𝑨 + 𝑩 + 𝑪

Order of addition does not matter


Vector components

It is customary to resolve a vector into components along mutually

perpendicular directions.


Determining vector components Given V and θ, we can find Vx, Vy

• In 2D, we can always write any vector as the sum of a

vector in the x-direction, and one in the y-direction.

• Given V, θ (magnitude, direction), we can find Vx and Vy

V =Vx +Vy

cos so cos VVV

Vx

x V

Vx

Vy

q sin so sin VV

V

Vy

y


x

y

V

Vtan

Given Vx and Vy, we can find V, θ

• Vx, Vy are the legs of the right triangle and are therefore perpendicular

• Vector 𝑽 as the hypotenuse.

• So, the magnitudes of the vectors satisfy the Pythagorean Theorem.

V

Vx

Vy

q

Vy

222

yx VVV

22

yx VVVV

x

y

V

V1tan so

x

y


Example 1

mV 10

• A vector is given by its magnitude and direction (V,)

• What is the x, y-component

of the vector?

axisxabove 30


Example 2

• A vector is given by its vector components:

• Write the vector in terms of magnitude and direction.

4,2 yx VV

47.42042 22 V

x

y

-1

4

2

-2

4yV

2xV

22

yx VVV

x

y

V

Vtan 2

2

4tan

magnitude

axisxfrom 11763180180

632tan 1

axisxfrom


Adding vectors by components

Given and , how can we find ?

21 VVV

V1

V2 V

V1x

V1y

V2x

V2y

V1 V2 V1 +V2

x

y

𝑽𝒙 =𝑽𝟏𝒙 + 𝑽𝟐𝒙

𝑽𝒚 =𝑽𝟏𝒚+𝑽𝟐𝒚

yx VVV ,

yx VVV 111 ,

yx VVV 222 ,

yyxx VVVV 2121 ,

Adding corresponding

components


Unit Vectors (1)

• As we said before, a vector has both magnitude and direction.

• Now, it’s time to simplify a notation of direction:

Let’s introduce unit vectors

vectorsunitasknownkji ˆ,ˆ,ˆ

x

y

z

𝒌

𝒋

𝒊

• They point along major axes of our

coordinate system

𝑖 = 𝑗 = 𝑘 =1

• Unit vectors have a magnitude of 1


Unit Vectors (2)

Writing a vector with unit vectors is equivalent to

multiplying each unit vector by a scalar

jVViVV yyxxˆ;ˆ

• If a vector has components:

• In unit vector notation, we write

3,4 yx VV

jiV ˆ3ˆ4

jViVV yxˆˆ

x

y

𝒋 𝒊

xV

),( yx VV

)3,4(

iVxˆ


Example: Vector Addition/Subtraction

A hiker traces her movement along a trail.

D1 = (2500m)i + (500m) j

D2 = (500m)i + (700m) j + (700m)k

D3 = (600m) j

D4 = -(500m)k

D = (3000m)i + (1800m) j + (200m)k

D =D1 +D2 +D3 +D4

= (2500m)i + (500m) j

+ (500m)i + (700m) j + (700m)k

+ (600m) j

- (500m)k

What is the hiker’s final displacement?

The first leg is a flat hike to the foot of the mountain: ----------------------------

On the second leg, she climbs the mountain:----------------------------------

On the third, she walks along a plateau: -----

Then she falls off a cliff: --------------------------


Relative Velocity

• So far we have just added/subtracted displacement vectors

• May find situations to add or subtract other types of

vectors, say velocity vectors

• Can only add or subtract the same type of vectors


Relative Velocity

5m/s

25m/s

• A train moves at 25 m/s relative to the ground

• Your velocity relative to the train is 5 m/s

• So your velocity relative to the ground is: 20 m/s

x

- +

+

-

+ +25 m/s – 5m/s=

//localhost/Users/Partha1/Documents/My Documents/Teaching/95.141/Wasserman-Fall10/Lectures/movies/Frame of Reference_1.mov


Boat in the river. Derivation of the relative velocity equation 3-15

𝑽𝑩𝑾 means “velocity of the boat relative to water”

𝑽𝑩𝑺 = 𝑽𝑩𝑾 + 𝑽𝑾𝑺

B Boat

W Water

S Shore


Relative Velocity - 2D

A boat’s speed in still water is 1.85m/s.

The river flows with a 1.20m/s current.

If we want to directly cross the stream at what upstream angle (see diagram) should the boat be pointed?

hypotenuse

oppositesin

River current

θ 𝑽𝑩𝑾

𝑽𝑾𝑺

𝑽𝑩𝑺

𝑽𝑩𝑾 means “velocity of the boat relative to water”

B Boat

W Water

S Shore

4.40)6486.0(sin 1

6486.085.1

20.1sin

𝑽𝑩𝑺 = 𝑽𝑩𝑾 + 𝑽𝑾𝑺 =1.85m/s

=1.20m/s


Summary

• Vectors

• Graphical Methods

• Addition and Subtraction

• Multiplication by a scalar

• Components

• Unit vectors

• Displacement & velocity vectors

• Relative Velocity


Thank you

See you on Wednesday

If two vectors are given

such that A + B = 0,

what can you say about

the magnitude and

direction of vectors

A and B?

A) same magnitude, but can be in any

direction

B) same magnitude, but must be in the same

direction

C) different magnitudes, but must be in the

same direction

D) same magnitude, but must be in opposite

directions

E) different magnitudes, but must be in

opposite directions

ConcepTest 1 Vectors I

If two vectors are given

such that A + B = 0, what

can you say about the

magnitude and direction of

vectors A and B?

A) same magnitude, but can be in any

direction

B) same magnitude, but must be in the same

direction

C) different magnitudes, but must be in the same

direction

D) same magnitude, but must be in opposite

directions

E) different magnitudes, but must be in

opposite directions

The magnitudes must be the same, but one vector must be pointing in the

opposite direction of the other in order for the sum to come out to zero. You

can prove this with the tip-to-tail method.

ConcepTest 1 Vectors I

𝑨 𝑩

Given that A + B = C, and

that lAl 2 + lBl 2 = lCl 2,

how are vectors A and B

oriented with respect to

each other?

1) they are perpendicular to each other

2) they are parallel and in the same direction

3) they are parallel but in the opposite

direction

4) they are at 45° to each other

5) they can be at any angle to each other

Note that the magnitudes of the vectors satisfy the Pythagorean Theorem.

This suggests that they form a right triangle, with vector C as the hypotenuse.

Thus, A and B are the legs of the right triangle and are therefore perpendicular.

ConcepTest 2 Vectors II

𝑨 𝑩

Pythagorean Theorem