Scalar & Vector tutorial

Transcription

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3 Scalar & Vector tutorial

4 scalar vector only magnitude, no direction both magnitude and direction 1-dimensional measurement of quantity not 1-dimensional time, mass, volume, speed temperature and so on force, velocity, momentum, acceleration, and so on

5 scalar time area mass work speed power length volume density temperature vectors lift drag force thrust weight velocity momentum acceleration displacement magnetic field

6 discussing now vs.

7 pseudovector = axial vector

8 pseudovector under an improper rotation (reflection) gains additional sign flip in 3-dimension, equal magnitude, opposite direction Geometrically, not mirror image

9 pseudovector not mirror image

10 polar vector = real vector

11 polar vector in 2-dimension has magnitude and direction in 2-dimension, can also have length and angle on reflection, matches with its mirror image

12 Polar Vector inherently possesses direction direction remains unchanged irrespective of the coordinate system chosen when its components are reversed, the vector obtained is different from the original vector

13 pseudovector (axial vector) polar vector (real vector) transforms like a vector, under proper rotation same gains additional sign flip, under improper rotation does not not mirror image on reflection matches with its mirror image, on reflection does not reverse sign when the coordinate axes are reversed reverses sign when the coordinate axes are reversed common examples: magnetic field, angular velocity common examples: velocity, force, momentum

14 in elementary mathematics representation of a vector specifying its length (size) specifying its endpoints in polar coordinates

15 polar coordinates = radial coordinate angular coordinate = polar angle they are defined in terms of Cartesian coordinates r θ x = r cos θ y = r sin θ r = radial distance from the origin θ = the counter-clockwise angle from the x-axis

16 Cartesian coordinate system right-hand rule Z X Y

17 Y the direction of a vector is represented as Y θ X θ the counter-clockwise angle of rotation X

18 easy measurement: magnitude and direction 12 km East 8 km North

19 What is radius vector? r θ The radius vector r = position vector from the origin to the current position dr r. = r.v dt v = velocity

20 The radius for circular motion = radius vector The radius vector locates the orbiting object The radius vector is not constant (constant size, direction changes) y O x O = origin at the center of the circle

21 vector cross product rule pseudovector X pseudovector = pseudovector polar vector X polar vector = pseudovector polar vector X pseudovector = polar vector

22 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

23 vector A under inversion original vector A their cross product remains invariant

24 discussing now

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26 equal vectors same magnitude same direction negative of a vector same magnitude opposite direction

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28 vector diagrams are illustrated by 2 points tip of the vector: destination of the arrow-head tail of the vector: the point of origin

29 expressing vectors using components a vector can be split into 2 orthogonal components A y A x by drawing an imaginary rectangle

30 expressing vectors using components A y θ A x 2 A = A + A x y 2 = tan 1 θ A A y x A x = A cos θ A y = A sin θ

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32 addition of vectors using Tip & Tail rule

33 addition of vectors using components A x B x A y B y A y B y A x B x

34 determining magnitude and direction of resultant vectors Pythagorean theorem Trigonometric functions

35 The Pythagorean Theorem opposite adjacent (hypotenuse) 2 = (opposite) 2 + (adjacent) 2

36 addition of 2 orthogonal vectors using Pythagorus theorem vector A vector B vector A 1 complete the right-handed triangle, 2 draw the hypotenuse = A + B

37 addition of 2 orthogonal vectors using Pythagorus theorem vector A vector B vector B vector A 1 2 complete the rectangle, draw the diagonal = A + B

38 vector B 90 vector A

39 Pythagorean theorem is NOT applicable to adding more than two vectors vectors that are not at 90 angle with each other

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41 rules of addition of vectors Parallelogram rule Trigonometric rule

42 addition of vectors Parallelogram rule vector B vector sum C = A + B scalar sum c a + b

43 addition of vectors Trigonometric rule θ 3 θ 2 θ 1 A sinθ B C 2 2 = = C = A + B 2AB cosθ3 1 sinθ2 sinθ3

44 subtraction of 2 vectors construct the negative of vector B construct the resultant C C = A B

45 A B subtraction of 2 vectors

46 useful links

47 useful links

48 useful links