vectors a vector quantity has both magnitude (size) and direction a scalar quantity only has size...

Vectors

A vector quantity has both magnitude (size) and direction

A scalar quantity only has size (i.e. temperature, time, energy, etc.)

Parts of a vector:

headtail

length – represents the magnitude

We can perform math operations with vectors!

Vector AdditionA motor boat is moving 15 km/hr relative to the water.

The river current is 10 km/hr downstream.

How fast does the boat go (relative to the shore)

upstream and downstream?

Boat Upstream Vector

Vector AdditionA motor boat is moving 15 km/hr relative to the water.

The river current is 10 km/hr downstream.

How fast does the boat go (relative to the shore)

upstream and downstream?

Boat Upstream Vector

Boat Downstream Vector

Vector AdditionA motor boat is moving 15 km/hr relative to the water.

The river current is 10 km/hr downstream.

How fast does the boat go (relative to the shore)

upstream and downstream?

Current Vector = 10 km/hr downstream

Boat Upstream Vector

Boat Downstream Vector

Boat Velocity UpstreamUpstream: Place vectors head to tail,

net result, 5 km/hr upstream

bas

Boat Velocity UpstreamUpstream: Place vectors head to tail,

Boat VelocityUpstream: Place vectors head to tail,

net result, 5 km/hr upstream

Start

Finish

Difference

Boat VelocityDownstream: Place vectors head to tail,

Boat VelocityDownstream: Place vectors head to tail,

net result,

Boat VelocityDownstream: Place vectors head to tail,

net result, 25 km/hr downstream

abba

Commutative law

bas

Forces On An AirplaneWhen will it fly?

Gravity

Propulsion

Net Force?

Forces On An AirplaneWhen will it fly?

Gravity

Propulsion

Net Force

Plane Dives to the Ground

Forces On An AirplaneWhen will it fly?

Gravity

Propulsion

Lift

Net Force?

FrictionWhen will it fly?

Gravity

Propulsion

Lift

Net Force = 0 up or down

Plane rolls along the runway like a car because of propulsion.

Forces On An AirplaneWhen will it fly?

Gravity

Propulsion

Lift

Net Force

Plane Flies as long as Lift > Gravity

FrictionWhen will it fly?

Gravity

Propulsion

Lift

Air Resistance

Net Force = 0

Equilibrium

FlightWhen will it fly?

Gravity

Propulsion

Lift

Air Resistance

Net Force

Plane Flies as long as Lift > Gravity

AND Propulsion > Air Resistance

Vector Components

A component of a vector is the projection of the vector on an axis

x

y

A

6A

Magnitude, size is:

We can write the vector A as the sum of an x-component and y-component:

yAxAA yx

Ax , Ay = the x and y components of the vector A

x hat and and y hat are the unit vectors

xA

yA

x

y

A

adjAx

oppAy

yAxAA yx

If we only know the mag. ofA, and the angle, it makeswith the x-axis, how do we find the x, and y components?

sinsin

coscos

AAA

A

hyp

opp

AAA

A

hyp

adj

xy

xx

x

y

A

adjAx

oppAy

yAxAA yx

If we only know the x andy components, how can we find the magnitude of A?

22yx AAA

This comes from Pythagorean’s theorem

GO TO HITT

Adding (and subtracting) vectors by components

yBxBB

yAxAA

yx

yx

Let’s say I have two vectors:I want to calculate the vectorsum of these vectors:

yBAxBABA yyxx

yxyBxBB

yxyAxAA

yx

yx

85

43

Let’s say the vectors have the following values:

x

y

yxyBxBB

yxyAxAA

yx

yx

85

43

A

B

yx

yx

yBAxBABA yyxx

122

8453

x

y

A

BOur result is consistentwith the graphical method!

What’s the magnitudeof our new vector?

2121481444

122 22

.

BA

x

y

A B+

How would you find the angle, , the vector makes with the y-axis?

yxBA

122

y

x

opp = 2

adj = 12

01 5961

61

122

.tantan adj

opp

GO TO HITT

Multiplying vectors by scalars:

yAaxAaAa

yAxAA

yx

yx

So if the vector A was:the scalar, a = 5 then the new vector:

yxA

43 and it was multiplied by

yx

yxAa

2015

4535

Scalar Product: (aka dot product):

cosbaba

mag. of amag. of b

angle betweenthe vectors

Scalar Product: (aka dot product):

cosbaba

vectors scalars

The dot product is the productof two quantities:(1) mag. of one vector(2) scalar component of the second vector along the direction of the first

zzyyxx

zyxzyx

bababa

zbybxbzayaxaba

)()(

Go To HITT

Vector Product (aka cross product)The vector product produces a new vector who’s magnitudeis given by:

sinbabac

The direction of the new vector is given by the, “right hand rule”

Mathematically, we can find the direction usingmatrix operations.

zbybxbb

zayaxaa

zyx

zyx

zyx

zyx

bbb

aaa

zyx

ba

The cross productis determined from three determinants

zyx

zyx

bbb

aaa

zyx

ba

The determinants are used to find the components of the vector

1st : Strike out the first column and first row!

3rd : Strike out the 2nd column and first row

4th : Cross multiply the four components,subtract, andmultiply by -1:

2nd : Cross multiply the four components – and subtract:

yzzy baba x - component

zyx

zyx

bbb

aaa

zyx

ba

xzzx baba y - component

zyx

zyx

bbb

aaa

zyx

ba

5th: Cross out the last column and first row

6th : Cross multiply and subtract four elements

xyyx baba z-component

So then the new vector will be:

zbabaybabaxbababac xyyxxzzxyzzy

We’ll look more at the scalar product when we talk about angular momentum.

Example:

yxb

yxa

24

32

zx

zyx

zyx

zyx

ba

1616

1240124

432240024322

024

032

Example:

yxb

yxa

24

32

z

zyx

zyx

zyx

ba

16

12400

432240022003

024

032

Notice the resultant vector is in the z – direction!