Vectors

VectorsThis is one of the most important

chapters in the course.

PowerPoint presentations are compiled from Walker 3rd Edition Instructor CD-ROM and Dr. Daniel Bullock’s own resources

Learning objectives• Scalars Versus Vectors• The Components of a Vector• Adding and Subtracting Vectors• Unit Vectors• Position, Displacement, Velocity, and Acceleration Vectors• Relative Motion

Scalars and Vectors• Scalar has only magnitude (size)

– 5 seconds (time)– 10 kg (mass)– 13.6 eV (energy)– 10 m/s (speed)

• Vector has both magnitude (size) and direction– 10 m/s North (velocity)– - 5 m/s2 (acceleration)– 4.5 Newtons 370 North of West (force)

Vectors• Vector has tail (beginning) and tip (end)• A position vector describes the location of a

point • Can resolve vector into perpendicular

components using a two-dimensional coordinate system:

tail

tip

Vectors• Length, angle, and components can be calculated from

each other using trigonometry:

AyA

hyp

oppsin

AxA

hyp

adjcos

x

y

A

A

adj

opptan

• Length (magnitude) of the vector A is given by the Pythagorean theorem

2222yx AAoppadj A

Vectors• Signs of vector components

Adding and Subtracting Vectors• Adding vectors graphically: Place the tail of the second

at the head of the first. The sum points from the tail of the first to the head of the last.

Adding Vectors Using Components:1. Find the components of each vector to be added.2. Add the x- and y-components separately.3. Find the resultant vector.

Adding and Subtracting Vectors• Subtracting Vectors: The negative of a

vector is a vector of the same magnitude pointing in the opposite direction. Here, D = A – B.

Unit Vectors• Unit vectors are dimensionless vectors of

unit length.• Multiplying unit vectors by scalars: the multiplier changes the length, and the sign indicates the direction.

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





Current Vector = 10 km/hr downstream


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,


net result,


net result, 25 km/hr downstream

abba

Commutative law

bas

Forces On An AirplaneWhen will it fly?

Gravity

Propulsion

Net Force?


Gravity

Propulsion

Net Force

Plane Dives to the Ground


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.


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

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

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

)()(

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!

Position, Displacement, Velocity, and Acceleration Vectors

Average velocity vector:

So vav is in the same

direction as Δr.

Position, Displacement, Velocity, and Acceleration Vector

• Average acceleration vector is in the direction of the change in velocity:

• Instantaneous velocity vector is tangent to the path:


• Velocity vector is always in the direction of motion; acceleration vector can point anywhere:


Relative Motion• The speed of the passenger with respect

to the ground depends on the relative directions of the passenger’s and train’s speeds:

Relative Motion

This also works in two dimensions:

Chapter 3 Summary

• Scalar: number, with appropriate units

• Vector: quantity with magnitude and direction

• Vector components: Ax = A cos θ, By = B sin θ

• Magnitude: A = (Ax2 + Ay

2)1/2

• Direction: θ = tan-1 (Ay / Ax)

• Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last

Chapter 3 Summary• Component method: add components of individual vectors, then find magnitude and direction• Unit vectors are dimensionless and of unit length• Position vector points from origin to location• Displacement vector points from original position to final position• Velocity vector points in direction of motion• Acceleration vector points in direction of change of motion• Relative motion: v13 = v12 + v23

Vectors

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