scalar & vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and...

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Page 2: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement
Page 3: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

Scalar & Vector tutorial

Page 4: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

scalar vector

time, mass, volume, speed temperature and so on

only magnitude, no direction both magnitude and direction

force, velocity, momentum, acceleration, and so on

1-dimensional measurement of quantity not 1-dimensional

Page 5: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

scalar

speed

volume

temperature

time

length

area

work

power

mass

density

vectors

weight

acceleration

lift

momentum

drag

force

velocity

thrust

displacement

magnetic field

Page 7: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

pseudovector = axial vector

Page 8: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

pseudovector

under an improper rotation (reflection)

gains additional sign flip

equal magnitude, opposite direction

not mirror image

in 3-dimension,

Geometrically,

Page 9: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

pseudovector

not mirror image

Page 10: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

polar vector = real vector

Page 11: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

polar vector

in 2-dimension

has magnitude and direction

can also have length and angle

matches with its mirror image

in 2-dimension,

on reflection,

Page 12: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

inherently possesses direction

Polar Vector

when its components are reversed, the vector obtained is

different from the original vector

direction remains unchanged irrespective of

the coordinate system chosen

Page 13: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

pseudovector (axial vector) polar vector (real vector)

gains additional sign flip, under improper rotation

transforms like a vector, under proper rotation same

does not

not mirror image on reflection

matches with its mirror image, on reflection

common examples: magnetic field, angular velocity

common examples: velocity, force, momentum

reverses sign when the coordinate axes are reversed

does not reverse sign when the coordinate axes are reversed

Page 14: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

representation of a vector

specifying its length (size) specifying its endpoints in polar coordinates

in elementary mathematics

Page 15: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

they are defined in terms of Cartesian coordinates

polar coordinates = radial coordinate

angular coordinate = polar angle

y = r sin θ x = r cos θ

r = radial distance from the origin

θ = the counter-clockwise angle from the x-axis

θ r

Page 17: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

the direction of a vector is represented as

the counter-clockwise

angle of rotation

θ X

Y

θ

X

Y

Page 19: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

θ r

What is radius vector ?

The radius vector r = “position vector” from the origin to the current position

dr dt

r. = r.v

v = velocity

Page 20: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

O = origin at the center of the circle

The radius for circular motion = radius vector

The radius vector locates the orbiting object

x

y

O

The radius vector is not constant

(constant size, direction changes)

Page 21: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

pseudovector X pseudovector = pseudovector

polar vector X polar vector = pseudovector

polar vector X pseudovector = polar vector

vector cross product rule

Page 22: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

vector triple product rule

polar vector X [ polar vector X polar vector ] = polar vector

pseudovector X [ polar vector X polar vector ] = pseudovector

polar vector X [ pseudovector X pseudovector ] = polar vector

pseudovector X [ pseudovector X pseudovector ] = pseudovector

pseudovector X [ polar vector X pseudovector ] = polar vector

polar vector X [ polar vector X pseudovector ] = pseudovector

Page 23: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

original vector A vector A under inversion

their cross product remains invariant

Page 25: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement
Page 26: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

equal vectors

same magnitude

same direction

negative of a vector

same magnitude

opposite direction

Page 27: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement
Page 28: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

vector diagrams are illustrated by 2 points

tail of the vector:

the point of origin

tip of the vector:

destination of the arrow-head

Page 29: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

expressing vectors using components

a vector can be split into 2 orthogonal components

Ax

A y

by drawing an imaginary rectangle

Page 30: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

expressing vectors using components

Ax = A cos θ

Ay = A sin θ

22yx AAA +=

x

y

AA1tan−=θ

Ax A y

θ

Page 31: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

&

Page 32: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

using “Tip & Tail rule”

addition of vectors

Page 33: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

using components addition of vectors

Ax

A y

Bx

B y

Ax

A y

B y

Bx

Page 35: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

The Pythagorean Theorem

adjacent

oppo

site

(hypotenuse)2 = (opposite)2 + (adjacent)2

Page 36: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

using Pythagorus’ theorem addition of 2 orthogonal vectors

vect

or B

vector A

vector A

1 complete the right-handed triangle,

draw the hypotenuse = A + B 2

Page 37: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

using Pythagorus’ theorem addition of 2 orthogonal vectors

vect

or B

vector A

vect

or B

vector A

1 2

complete the rectangle,

draw the diagonal = A + B

Page 38: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

vect

or B

vector A

90°

Page 39: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

Pythagorean theorem is NOT applicable to

vectors that are not at 90° angle with each other

adding more than two vectors

Page 40: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement
Page 41: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

rules of addition of vectors

Parallelogram rule

Trigonometric rule

Page 42: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

addition of vectors

Parallelogram rule

vector B

vector sum C = A + B

scalar sum c ≠ a + b

Page 43: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

addition of vectors

Trigonometric rule

θ2 θ1

θ3

321 sinsinsin θθθCBA

== 322 cos2 θABBAC −+=

Page 44: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

subtraction of 2 vectors

construct the negative of vector B

construct the resultant C

C = A – B

Page 45: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

subtraction of 2 vectors

A – B

Page 46: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

useful links

http://arc.aiaa.org/doi/abs/10.2514/8.4525?journalCode=jars

http://www.eolss.net/sample-chapters/c05/e6-10-03-00.pdf

http://phrontistery.info/index.html

http://physics.stackexchange.com/

http://www.physicsclassroom.com/class/1DKin/Lesson-1/

http://www.efm.leeds.ac.uk/CIVE/CIVE1140/section01/

http://hydrorocketeers.blogspot.dk/2011_06_01_archive.html

http://whatis.techtarget.com/definition/

http://physics.info/

http://study.com/academy/lesson/

http://physics.tutorvista.com/motion/

http://www.wikidiff.com/

http://hyperphysics.phy-astr.gsu.edu/hbase/

Page 47: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

http://www.physicsclassroom.com/class/circles/

http://physics.weber.edu/

http://physics.about.com/od/classicalmechanics/

http://theory.uwinnipeg.ca/physics/

http://physics.oregonstate.edu/~stetza/ph407H/Quantum.pdf

https://www.aip.org/history/exhibits/heisenberg/p08.htm

http://plato.stanford.edu/entries/qt-uncertainty/

http://www.begellhouse.com/books/4799b1c911961558.html

http://functionspace.com/articles/8/Introduction-to-Classical-Mechanics--Statics-Kinematics-Dynamics

https://bayanbox.ir/view/7764531208313247331/Kleppner-D.-Kolenkow-R.J.-Introduction-to-Mechanics-2014.pdf

https://owlcation.com/stem/Physics-Definition-and-Branches

useful links

http://plato.stanford.edu/entries/equivME/

http://mathworld.wolfram.com/topics/

Page 48: Scalar & Vector tutorial · 7/26/2018  · only magnitude, no direction . both magnitude and direction . force, velocity, momentum, acceleration, and so on . 1-dimensional measurement

http://www.space.com/17661-theory-general-relativity.html

https://en.wikipedia.org/wiki/General_relativity

http://www.physicsoftheuniverse.com/topics_relativity_general.html

http://www.dummies.com/education/science/physics/einsteins-general-relativity-theory-gravity-as-geometry/

http://www.dummies.com/education/science/physics/einsteins-special-relativity/

http://physics.oregonstate.edu/~stetza/ph407H/Quantum.pdf

http://whatis.techtarget.com/definition/quantum-theory

useful links

http://ircamera.as.arizona.edu/NatSci102/NatSci102/lectures/physicallaws.htm

http://physics.info/photoelectric/

https://www.boundless.com/physics/textbooks/boundless-physics-textbook/

http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/ltrans.html