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