PHYS 207.013

Chapter 3

vectors

Instructor: Dr. Bianco

TAs: Joey Betz; Lily Padlow

University of Delaware - Spring 2021

https://slides.com/federicabianco/phys20713_3

H&R CH3 vectors

0. demo

vectors and scalars

F

H&R CH3 vectors

vectors and scalars

we live in a 3D physical space so quantities that describe how we move within it are typically vectors (but not only!)

1. definition

a vector is a physical quantity with 2 properties: magnitude and direction

H&R CH3 vectors

vectors and scalars

a vector is a physical quantity with 2 properties: magnitude and direction

2. notation

\vec{a}

We draw vectors as arrows with appropriate length and direction specified with respect to a coordinate system
The length of a vector when drawn corresponds to its magnitude

H&R CH3 vectors

vectors and scalars

a vector is a physical quantity with 2 properties: magnitude and direction

2. notation

\vec{a}

We draw an arrow above a symbol to indicate its a vector
We sometimes (the book does) use bold font to indicate a symbols refers to a vector
To refer to the magnitude of the vector we use the absolute value notation
We sometimes just drop the arrow to refer to the magnitude of the vector

\vec{a}

\mathbf{a}

|\vec{a}|

{a}

vectors and scalars

a vector is a physical quantity with 2 properties: magnitude and direction

2. notation

\vec{a}

We draw an arrow above a symbol to indicate its a vector
We sometimes (the book does) use bold font to indicate a symbols refers to a vector
To refer to the magnitude of the vector we use the absolute value notation
We sometimes just drop the arrow to refer to the magnitude of the vector
The direction of the vector is quantified by the angle it makes with the coordinates' axis
Angles are usually referred to by a greek letter

\vec{a}

\mathbf{a}

|\vec{a}|

\theta

{\theta}

{a}

vectors and scalars

length

time

displacement

mass

pressure

acceleration

density

temperature

force

velocity

momentum

H&R CH3 vectors

speed

energy

vectors and scalars

length

time

displacement

mass

pressure

density

temperature

force

velocity

momentum

H&R CH3 vectors

speed

acceleration

energy

vector operations

\vec{a}

\vec{b}

\vec{a} + \vec{b}

H&R CH3 vectors

\vec{a}

\vec{b}

\vec{a} + \vec{b}

vector sum: the resultant vector from a sum of a series of vectors is the vector that joins the starting point with the ending point

y

the point may be in displacement space, or in velocity space, or in any other vector space

vector operations

H&R CH3 vectors

vy

vx

x

\vec{a}

\vec{b}

\vec{a} + \vec{b}

vector sum: the resultant vector from a sum of a series of vectors is the vector that joins the starting point with the ending point

If two vectors start at the same point their sum is the vector resulting by drawing a parallelogram and joining the opposite corners of it

x

y

vector operations

H&R CH3 vectors

vector sum: the resultant vector from a sum of a series of vectors is the vector that joins the starting point with the ending point

\vec{a}

\vec{b}

\vec{a} + \vec{b}

vector operations

H&R CH3 vectors

x

y

Properties

Commutative:

\vec{a} + \vec{b} = \vec{b} +\vec{a}

vector sum: the resultant vector from a sum of a series of vectors is the vector that joins the starting point with the ending point

Properties

Commutative:

Associative

\vec{a} + \vec{b} = \vec{b} +\vec{a}

(\vec{a} + \vec{b}) + \vec{c} = \vec{a} +(\vec{b} + \vec{c})

\vec{a}

\vec{b}

\vec{a} + \vec{b} + \vec{c}

\vec{c}

vector operations

H&R CH3 vectors

x

y

vector sum: the resultant vector from a sum of a series of vectors is the vector that joins the starting point with the ending point

-\vec{a}

\vec{a}

vector subtraction: the sign of a vector relates to its direction

\vec{a} - \vec{b} = \vec{a} +( -\vec{b})

\vec{a} - \vec{a} = 0

vector operations

H&R CH3 vectors

x

y

Properties

Commutative:

Associative

\vec{a} + \vec{b} = \vec{b} +\vec{a}

(\vec{a} + \vec{b}) + \vec{c} = \vec{a} +(\vec{b} + \vec{c})

vector sum: the resultant vector from a sum of a series of vectors is the vector that joins the starting point with the ending point

\vec{a}

\vec{b}

\vec{a} + \vec{b}

vector operations

vector sum: the resultant vector from a sum of a series of vectors is the vector that joins the starting point with the ending point

-\vec{b}

\vec{a} - \vec{b}

x

y

vector subtraction: the sign of a vector relates to its direction

\vec{a} - \vec{b} = \vec{a} +( -\vec{b})

\vec{a} - \vec{a} = 0

Properties

Commutative:

Associative

\vec{a} + \vec{b} = \vec{b} +\vec{a}

(\vec{a} + \vec{b}) + \vec{c} = \vec{a} +(\vec{b} + \vec{c})

H&R CH3 vectors

x

y

\vec{c}

c_x

c_y

vector component: the component of a vector is its projection along an axis

\theta

vector operations

\vec{c} = c_x\hat{i} + c_y{\hat{j}}

c_x = c \cos(\theta)

c_y = c \sin(\theta)

H&R CH3 vectors

|\vec{c}| = \sqrt{c_x^2 + c_y^2}

\hat{j}

\hat{i}

\hat{k}

MATH REVIEW:

vector math

vector sum: the sum of 2 vectors reduces to the sum of 2 scalars for each component of the vector

x

\vec{a}

\vec{b}

\vec{c} = \vec{a} + \vec{b}

c_x = a_x + b_x =

c_y = a_y + b_y

vector component: the component of a vector is its projection along an axis

a \cos(\alpha) + b \cos(\beta)

a \sin(\alpha) + b \sin(\beta)

H&R CH3 vectors

\alpha

\beta

MATH REVIEW:

vector math

vector sum: the sum of 2 vectors reduces to the sum of 2 scalars for each component of the vector

x

y

\vec{a}

\vec{b}

\vec{c} = \vec{a} + \vec{b}

c_x = a_x + b_x =

c_y = a_y + b_y

vector component: the component of a vector is its projection along an axis

a \cos(\alpha) + b \cos(\beta)

a \sin(\alpha) + b \sin(\beta)

\alpha

\beta

H&R CH3 vectors

MATH REVIEW:

trigonometry

vector component: the component of a vector is its projection along an axis which you can derive using trigonometry

H&R CH3 vectors

https://www.geogebra.org/m/aavMVjyK

MATH REVIEW:

trigonometry

vector component: the component of a vector is its projection along an axis which you can derive using trigonometry

H&R CH3 vectors

https://www.geogebra.org/m/aavMVjyK

things to notice:

the cosine is measure along the x axis

the sine is measure along the y axis

the angles repeat identically after you go 360 degrees around

MATH REVIEW:

trigonometry

vector component: the component of a vector is its projection along an axis which you can derive using trigonometry

H&R CH3 vectors

https://www.geogebra.org/m/aavMVjyK

things to notice:

the cosine is measure along the x axis

the sine is measure along the y axis

the angles repeat identically after you go 360 degrees around

each quadrant has a specific sign for the cosine/sine

Top Right

Bottom Right

Bottom Left

Top Left

MATH REVIEW:

trigonometry

H&R CH3 vectors

the sine and cosine of some common angles

\mathrm{at}~\theta = (\pm \pi)

\cos{\theta} =1

\sin{\theta}=0

}

\mathrm{at}~\theta = (\pm \frac{\pi}{2})

\cos{\theta} =0

\sin{\theta}=1

}

H&R CH1 measuring things - including length

MATH REVIEW:

trigonometry

\pi~ radians = 180^o

60^o =

60^o = 60^o \cdot \frac{\pi}{180^o} =

3

60^o = 60^o \cdot \frac{\pi}{180^o} = \frac{\pi}{3}~radians

MATH REVIEW:

trigonometry

\cos(\theta)^2 + \sin(\theta)^2 = 1

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha)

\sin(\alpha-\beta) = \sin(\alpha)\cos(\beta) - \sin(\beta)\cos(\alpha)

\cos(\alpha+\beta) = \cos(\alpha)\cos(\beta) - \sin(\beta)\sin(\alpha)

r

\alpha

\cos(\alpha-\beta) = \cos(\alpha)\cos(\beta) + \sin(\beta)\sin(\alpha)

r \sin(\alpha)

r \cos(\alpha)

H&R CH3 vectors

useful trig tricks!

MATH REVIEW:

trigonometry

\cos(\theta)^2 + \sin(\theta)^2 = 1

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

r

\alpha

r \sin(\alpha)

r \cos(\alpha)

H&R CH3 vectors

useful trig tricks!

\theta = \tan^{-1}(\tan{(\theta)}) = \arctan{(\tan{(\theta)})}

MATH REVIEW:

vector math

vector sum: the sum of 2 vectors reduces to the sum of 2 scalars for each component of the vector

x

\vec{a}

\vec{b}

\vec{c} = \vec{a} + \vec{b}

c_x = a_x + b_x =

c_y = a_y + b_y

vector component: the component of a vector is its projection along an axis

a \cos(\alpha) + b \cos(\beta)

a \sin(\alpha) + b \sin(\beta)

H&R CH3 vectors

\alpha

\beta

MATH REVIEW:

vector math

vector sum: the sum of 2 vectors reduces to the sum of 2 scalars for each component of the vector

vector component: the component of a vector is its projection along an axis

what sign does cx have?

what sign does cy have?

\vec{c} = \vec{a} + \vec{b}

https://phet.colorado.edu/sims/html/vector-addition/latest/vector-addition_en.html

vector products

Multiply vector by scalar
Multiply vector by vector

2.1- Dot product

2.2- Vector product

\vec{a} \cdot \vec{b} => \mathrm{scalar}

\vec{a} \times \vec{b} => \mathrm{vector}

H&R CH3 vectors

there are 3 kinds of products that involve vectors

vector products

Multiply vector by scalar

c ~ \vec{a}

the magnitude of changes by a factor c.

the direction is unchanged

H&R CH3 vectors

the result is a vector

\vec{a}

vector products

Multiply vector by vector : Dot product

\vec{a} \cdot \vec{b}

the result is a scalar

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \phi

\vec{a}

\phi

x

\vec{b}

y

H&R CH3 vectors

results is the product of the magnitudes times the product of the cosine of the angle between the vectors

vector products

2. Multiply vector by vector : Dot product

the result is a scalar

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \phi

\vec{a}

\phi

x

\vec{b}

r=1

\phi

y

H&R CH3 vectors

results is the product of the magnitudes times the product of the cosine of the angle between the vectors

\vec{a} \cdot \vec{b}

the result is a scalar

vector products

Multiply vector by vector : Cross product

\vec{a} \times \vec{b}

the result is a vector

\vec{a} \times \vec{b} = \vec{c}

\vec{a}

\phi

y

x

\vec{b}

result is a vector

whose magnitude is the product of the magnitudes times the sin of the angle between the vectors

whose direction is given by the right hand rule

|\vec{c}| = |\vec{a}| |\vec{b}| \sin \phi

H&R CH3 vectors

vector products

Multiply vector by vector : Cross product

the result is a vector

\vec{a} \times \vec{b} = \vec{c}

\vec{a}

\phi

y

x

\vec{b}

|\vec{c}| = |\vec{a}| |\vec{b}| \sin \phi

direction perpendicular to the plane of the 2 vectors, following the right hand rule

H&R CH3 vectors

\hat{j}

\hat{i}

\hat{k}

\vec{a} \times \vec{b}

the result is a vector

vector products

Multiply vector by vector : Cross product

the result is a vector

\vec{a} \times \vec{b} = \vec{c}

\vec{a}

\phi

y

x

\vec{b}

|\vec{c}| = |\vec{a}| |\vec{b}| \sin \phi

direction perpendicular to the plane of the 2 vectors, following the right hand rule

H&R CH3 vectors

\hat{j}

\hat{i}

\hat{k}

\vec{a} \times \vec{b}

the result is a vector

vector products

Multiply vector by vector : Cross product

the result is a vector

\vec{a} \times \vec{b} = \vec{c}

\vec{a}

\phi

y

x

\vec{b}

direction perpendicular to the plane of the 2 vectors, following the right hand rule

let's look at the components:

H&R CH3 vectors

{a}_x\hat{i} \times {b}_x \hat{i}= a_xb_x\times (\hat{i}\times\hat{i}) =

{a}_x\hat{i} \times {b}_x \hat{i}= a_xb_x\times (\hat{i}\times\hat{i}) =0

{a}_x\hat{i} \times {b}_y \hat{j}= a_xb_y\times (\hat{i}\times\hat{j}) =a_xb_y\vec{k}

{a}_x\hat{i} \times {b}_y \hat{j}= a_xb_y\times (\hat{i}\times\hat{j}) =

\hat{j}

\hat{i}

\hat{k}

-{a}_x\hat{i} \times {b}_y \hat{j}= a_xb_y\times (\hat{-i}\times\hat{j}) = - a_xb_y \hat{k}

-{a}_x\hat{i} \times {b}_y \hat{j}= a_xb_y\times (\hat{-i}\times\hat{j}) =

\vec{a}_x

\vec{b}_x

\vec{a}_y

\vec{b}_y

KEY POINTS:

vectors can be summed geometrically with the parallelogram rule

vectors can be summed algebraically by splitting them into their components

the components of a vector are its projections on the coordinate axes

vectors have 2 properties: magnitude and direction

the direction of a vector ie measured as the angle it makes counterclockwise with the x axis

the projections are found with trigonometry

H&R CH3 vectors

KEY POINTS:

cross product of 2 vectors produces a vector with magnitude that points perpendicularly to the plane of the 2 vectors according to the right hand rule

vector multiplication with a scalar changes the magnitude of the vector

dot product of vectors produces a scalar with magnitude

\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \phi

|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \phi

| c \vec{a} |= c|\vec{a}|

H&R CH3 vectors

KEY POINTS:

the tangent of an angle is the ration of sin/cosin and this relation can be inverted to measure angles

the cosine of an angle is its projection on the x axis for a vector of unit magnitude

the sine of an angle is its projection on the y axis for a vector of unit magnitude

the tangent of an angle is the ration of sin/cosin and this relation can be inverted to measure angles

the sum of the square of cosine and sine is equal to 1

\cos(\theta)^2 + \sin(\theta)^2 = 1

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

H&R CH3 vectors