scalar & vector tutorial · 7/26/2018 · only magnitude, no direction . both magnitude and...
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Scalar & Vector tutorial
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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
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scalar
speed
volume
temperature
time
length
area
work
power
mass
density
vectors
weight
acceleration
lift
momentum
drag
force
velocity
thrust
displacement
magnetic field
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pseudovector = axial vector
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pseudovector
under an improper rotation (reflection)
gains additional sign flip
equal magnitude, opposite direction
not mirror image
in 3-dimension,
Geometrically,
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pseudovector
not mirror image
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polar vector = real vector
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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,
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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
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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
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representation of a vector
specifying its length (size) specifying its endpoints in polar coordinates
in elementary mathematics
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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
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Cartesian coordinate system right-hand rule
X
Y
Z
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the direction of a vector is represented as
the counter-clockwise
angle of rotation
θ X
Y
θ
X
Y
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12 km East 8
km N
orth
easy measurement: magnitude and direction
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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
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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)
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pseudovector X pseudovector = pseudovector
polar vector X polar vector = pseudovector
polar vector X pseudovector = polar vector
vector cross product rule
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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
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original vector A vector A under inversion
their cross product remains invariant
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discussing now………………
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equal vectors
same magnitude
same direction
negative of a vector
same magnitude
opposite direction
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vector diagrams are illustrated by 2 points
tail of the vector:
the point of origin
tip of the vector:
destination of the arrow-head
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expressing vectors using components
a vector can be split into 2 orthogonal components
Ax
A y
by drawing an imaginary rectangle
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expressing vectors using components
Ax = A cos θ
Ay = A sin θ
22yx AAA +=
x
y
AA1tan−=θ
Ax A y
θ
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&
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using “Tip & Tail rule”
addition of vectors
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using components addition of vectors
Ax
A y
Bx
B y
Ax
A y
B y
Bx
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determining magnitude and direction of resultant vectors
Pythagorean theorem
Trigonometric functions
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The Pythagorean Theorem
adjacent
oppo
site
(hypotenuse)2 = (opposite)2 + (adjacent)2
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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
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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
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vect
or B
vector A
90°
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Pythagorean theorem is NOT applicable to
vectors that are not at 90° angle with each other
adding more than two vectors
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rules of addition of vectors
Parallelogram rule
Trigonometric rule
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addition of vectors
Parallelogram rule
vector B
vector sum C = A + B
scalar sum c ≠ a + b
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addition of vectors
Trigonometric rule
θ2 θ1
θ3
321 sinsinsin θθθCBA
== 322 cos2 θABBAC −+=
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subtraction of 2 vectors
construct the negative of vector B
construct the resultant C
C = A – B
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subtraction of 2 vectors
A – B
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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/
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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/
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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