Vectors and scalars - PowerPoint Presentation

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Vectors and scalars - PPT Presentation

A scalar quantity can be described by a single number with some meaningful unit 4 oranges 20 miles 5 mileshour 10 Joules of energy 9 Volts Vectors and scalars A scalar quantity ID: 437482

vector vectors amp components vectors vector components amp sons wiley john 2014 reserved rights direction product magnitude scalar unit

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Slide1

Vectors and scalars

scalar

quantity

can be described by a

single number

with some meaningful

unit

4 oranges

20 miles

5 miles/hour

10 Joules of energy

9 Volts Slide2

Vectors and scalars

scalar

quantity

can be described by a

single number with some meaningful

unit

vector

quantity

has a

magnitude

and a

direction

in space,

as well as some meaningful

unit

5 miles/hour North

Newtons

in the “x direction”

50 Volts/meter downSlide3

3-1

Vectors and Their Components

The simplest example is a

displacement vector

If a particle changes position from A to B, we represent this by a vector arrow pointing from A to B

Figure 3-1

In (a) we see that all three arrows have the same magnitude and direction: they are identical displacement vectors.

In (b) we see that all three paths correspond to the same displacement vector. The vector tells us nothing about the actual path that was taken between A and B.

Vectors and scalars

scalar

quantity

can be described by a

single number with some meaningful unitA vector quantity has a magnitude and a direction in space, as well as some meaningful unit.

To establish the

direction, you MUST first have a coordinate system! Standard x-y Cartesian coordinates commonCompass directions (N-E-S-W)Slide5

Drawing vectors

Draw a vector as a line with an arrowhead at its tip.

The

length

of the line shows the vector’s

magnitude

The direction of the line shows the vector’s direction

relative to a coordinate system (that should be indicated!)

5 m/sec at

30 degrees from the x axis towards y in the xy planeSlide6

Drawing vectors

Vectors can be identical in magnitude, direction, and units, but start from different places…Slide7

Drawing vectors

Negative

vectors refer to direction relative to some standard coordinate already established – not to magnitude.Slide8

Adding two vectors graphically

Two vectors may be added graphically using either the

head-to-tail

method or the

parallelogram

method.

Slide9

Adding two vectors graphically

Two vectors may be added graphically using either the

head-to-tail

method or the

parallelogram

method.Slide10

Adding two vectors graphicallySlide11

3-1

Vectors and Their Components

The

vector sum

, or

resultantIs the result of performing vector addition

Represents the net displacement of two or more displacement vectors

Can be added graphically as shown:

Figure 3-2

Eq. (3-1)

3-1

Vectors and Their Components

Vector addition is

commutative

We can add vectors in any order

Eq. (3-2)

Figure (3

-3)

3-1

Vectors and Their Components

Vector addition is

associative

We can group vector addition however we like

Eq. (3-3)

Figure (3

-4)

3-1

Vectors and Their Components

A negative sign reverses vector direction

We use this to define vector subtraction

Eq. (3-4)

Figure (3

-5)

Figure (3

-6)

3-1

Vectors and Their Components

These rules hold for all vectors, whether they represent displacement, velocity, etc.

Only vectors of the same kind can be added

(

distance

) + (

distance) makes sense

3-1

Vectors and Their Components

These rules hold for all vectors, whether they represent displacement, velocity, etc.

Only vectors of the same kind can be added

(

distance

) + (

distance) makes sense

(distance) + (velocity) does notAnswer:

(a) 3 m + 4 m =

7 m

(b) 4 m - 3 m =

1 m

Adding more than two vectors graphically

To add several vectors, use the head-to-tail method.

The vectors can be added in any order.Slide18

Adding more than two vectors graphically—Figure 1.13

To add several vectors, use the head-to-tail method.

The vectors can be added in any order.Slide19

Subtracting vectors

Reverse direction, and add normally head-to-tail…Slide20

Subtracting vectorsSlide21

Multiplying a vector by a scalar

is a scalar, the product

A has magnitude |c|A.

Slide22

Addition of two vectors at right angles

First add vectors graphically.

Use trigonometry to find magnitude & direction of sum.Slide23

Addition of two vectors at right angles

Displacement (D) =

√

(1.00

+ 2.00

) = 2.24 km

Direction f = tan-1(2.00/1.00) = 63.4º East of NorthSlide24

Note how the final answer has THREE things!

Answer:

2.24

km at 63.4 degrees East of North

Magnitude

(with correct sig. figs!)Slide25

Note how the final answer has THREE things!

Answer: 2.24

at 63.4 degrees East of North

Magnitude (with correct sig. figs!)

UnitsSlide26

Note how the final answer has THREE things!

Answer: 2.24 km at

63.4 degrees East of North

Magnitude (with correct sig. figs!)

Units

DirectionSlide27

Components of a vector

Represent any vector by an

-component

and a

y-component Ay.Use trigonometry to find the components of a vector:

Ax = Acos θ and Ay = Asin θ, where

is measured from the +

axis toward the +

axis. Slide28

Positive and negative components

The components of a vector can be positive or negative numbers.Slide29

Finding components

We can calculate the components of a vector from its magnitude and direction.

Slide30

3-1

Vectors and Their Components

Components in two dimensions can be found by:

Where

is the angle the vector makes with the positive

axis, and

a is the vector lengthThe length and angle can also be found if the components are known

3-1

Vectors and Their Components

In the three dimensional case we need more components to specify a vector

(a,

) or (ax,ay,az)

3-1

Vectors and Their Components

In the three dimensional case we need more components to specify a vector

(a,

) or (ax,ay,az)

Answer: choices

(c), (d), and (f)

show the components properly arranged to form the vector

Calculations using components

We can use the components of a vector to find its magnitude and direction:

can use the components of a set of vectors to find the components of their sum:Slide34

Adding vectors using their componentsSlide35

Unit vectors

unit vector

has a magnitude of 1 with no units.

The unit vector

points in the +

-direction, points in the +y-direction, and points in the +z-direction.Any vector can be expressed in terms of its components as

+ A

z .

Slide36

3-2

Unit Vectors, Adding Vectors by Components

unit vector

Has magnitude 1

Has a particular direction

Lacks both dimension and unit

Is labeled with a hat:

^We use a right-handed coordinate system

Remains right-handed when rotated

3-2

Unit Vectors, Adding Vectors by Components

The quantities

i and ayj

are vector components

The quantities

ax and

alone are

scalar

components

Vectors can be added using components

Eq. (3-9)

→

3-2

Unit Vectors, Adding Vectors by Components

To subtract two vectors, we subtract components

Unit Vectors, Adding Vectors by Components

Eq. (3-13)

3-2

Unit Vectors, Adding Vectors by Components

To subtract two vectors, we subtract components

Unit Vectors, Adding Vectors by Components

Answer:

(a) positive, positive (b) positive, negative

Eq. (3-13)

3-3

Multiplying Vectors

Multiplying two vectors: the

scalar product

Also called the

dot product

Results in a scalar, where

a and b are magnitudes and φ

is the angle between the directions of the two vectors:The commutative law applies, and we can do the dot product in component form

Eq. (3-20)

Eq. (3-22)

Eq. (3-23)

3-3

Multiplying Vectors

A dot product is: the product of the magnitude of one vector times the scalar component of the other vector in the direction of the first vector

Eq. (3-21)

Figure (3-18)

Either projection of one vector onto the other can be used

To multiply a vector by the projection, multiply the magnitudes

3-3

Multiplying Vectors

Answer:

(a) 90 degrees (b) 0 degrees (c) 180 degrees

The scalar product

The scalar product of two vectors (the “dot product”) is

· B =

ABcos

fSlide44

The scalar product

The scalar product of two vectors (the “dot product”) is

· B =

ABcos

fSlide45

The scalar product

The scalar product of two vectors (the “dot product”) is

· B =

ABcos

Useful for

Work (energy) required or released as force is applied over a distance (4A)Flux of Electric and Magnetic fields moving through surfaces and volumes in space (4B)Slide46

Calculating a scalar product

By components, A · B =

AyBy + AzB

z Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130°Slide47

Calculating a scalar product

By components, A · B =

AyBy + AzB

z Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130° Ax = 4.00 cos 53 = 2.407 Ay = 4.00 sin 53 = 3.195 Bx = 5.00 cos 130 = -3.214 By = 5.00 sin 130 = 3.830AxBx + Ay

= 4.50 meters

· B =

ABcos

= (4.00)(5.00) cos(130-53) = 4.50 meters2Slide48

The vector product

The vector product (“cross product”) of two vectors has magnitude

and the

right-hand rule

gives its direction. Slide49

3-3

Cross Products

The

cross product

of two vectors with magnitudes a & b, separated by angle φ,

produces a vector with magnitude:

And a direction perpendicular to both original vectors

Direction is determined by the right-hand rulePlace vectors tail-to-tail, sweep fingers from the first to the second, and thumb points in the direction of the resultant vector

Eq. (3-24)

3-3

Multiplying Vectors

The upper shows vector

cross vector

b, the lower shows vector b cross vector a

Figure (3-19)

The vector product

The vector product (“cross product”)

A x B

of two vectors is a vector

Magnitude = AB sin

fDirection = orthogonal (perpendicular) to A and B, using the “Right Hand Rule”

A x B

zSlide52

3-3

Multiplying Vectors

The cross product is not commutative

To evaluate, we distribute over components:

Therefore, by expanding (3-26):

Eq. (3-25)

Eq. (3-26)

Eq. (3-27)

3-3

Multiplying Vectors

3-3

Multiplying Vectors

Answer:

(a) 0 degrees (b) 90 degrees

The vector cross product

The cross product of two vectors is

x B (with magnitude

ABsin

)

Useful for Torque from a force applied at a distance away from an axle or axis of rotation (4A)Calculating dipole moments and forces from Magnetic Fields on moving charges (4B)Slide56

3-1

Vectors and Their Components

Angles may be measured in degrees or radians

Recall that a full circle is 360

, or 2

π rad

Know the three basic trigonometric functions

Figure (3-11)

3-2

Unit Vectors, Adding Vectors by Components

Vectors are independent of the coordinate system used to measure them

We can rotate the coordinate system, without rotating the vector, and the vector remains the same