Which came first: Vector Product or Torque? by Antonia Katsinos.

Which came first: Vector Product or Torque?

by Antonia Katsinos

VECTOR ALGEBRA vs. TORQUECROSS PRODUCT

1844 – Hermann Grassman

1843 – William Rowan Hamilton

1880-1884 Josiah Willard Gibbs

1880 - Oliver Heaviside

TORQUE

Rotational/angular force 300 BC – Archimedes

work of on levers

1687 - Newton’s Second Law of Rotation

1922 - Jack Johnson patented the

wrench

Torque = Force applied x

lever arm

TWO TRADITIONSTHE STUDY OF NUMBERS Natural numbers Negatives numbers,

zero, fractions & irrational #s

solving the equation x^2 +1 = 0 led to complex numbers

Their geometrical representation in space led to vector analysis

PHYSICAL PHENOMENA need to describe with

magnitude and direction such as velocity

plus the need of geometry to approach physical problems

together brought forward to concept of a vector

HERMAN GRASSMANN(April 15, 1809 – Sept. 26, 1877)

1832 was a high school math teacher at the Gymnasium in Germany

while teaching continued his father’s research on the concept of product in geometry

Recognized a relationship between sums and products; whether you multiply the sum of two displacements by a third displacement lying in the same plane, or the individual terms by the same displacement and add the products with due regard for their positive and negative values, the same result is obtained

1840 during the writing of further examinations he realized that he could apply the vector methods he had been researching to the essay topic of the theory of the tides

his system is closer to our present day vector algebra, but unrecognized because of his obscurity and his books’ unreadability

WILLIAM ROWAN HAMILTON(August 4, 1805 – September 2, 1865)

one interest was the relationship between complex numbers and geometry

sought an algebra of complex numbers that bears the same relationship with 3D geometry

unexpectedly his search ended on 16 October 1843 when he realized that the appropriate algebra was not a triplet algebra but a 4D algebra of what he called "quaternions“

many mathematical terms in common use today - including scalar and vector - were introduced by Hamilton, as he developed the theory of quaternions

QUATERNIONS A quaternion is a 4D complex number that

is of the form q = w + xi + yj + zk, where i, j and k are all different square roots of -1.

The quaternion can be regarded as an object composed of a scalar part, w, which is a real number, and a vector part, xi + yj + zk.

the vector part may be represented, in magnitude and direction, by a line joining two points in 3D space.

JOSIAH WILLARD GIBBS(February 11, 1839 – April 18, 1903)

from 1880 to 1884, Gibbs combined the ideas of two mathematicians, the quaternions of William Rowan Hamilton and the exterior algebra of Herman Grassmann to obtain vector analysis

Gibbs designed vector analysis to clarify and advance mathematical physics

From 1882 to 1889, Gibbs refined his vector analysis, wrote on optics, and developed a new electrical theory of light

OLIVER HEAVISIDE(May 18, 1850 – February 3, 1925)

both Gibbs and Heaviside arrived at identical systems by modifying Hamilton’s quaternions algebra, working independently of each other

while both Gibbs and Heaviside started with Hamilton’s methods, the system they both arrived at was closer to Grassmann’s in structure

improved vector terminology promoted the use of vectors in his 1893

book “Electromagnetic Theory”

Vector Analysis was created in the late 19th century when

Hamilton’s quaternion system was adapted to the needs of physics by Clifford, Tait, Maxwell, Heaviside, Gibbs and others

many earlier results obtained by Lagrange, Gauss, Green and others on hydrodynamics, sound and electricity, were then re-expressed in terms of vector analysis.

many of the vector analysis topics are now taught in courses on the “calculus on manifolds.”

BASIC VECTOR OPERATIONS Both a magnitude and a direction must be specified

for a vector quantity Any number of vector quantities of the same type

(i.e., same units) can be combined by basic vector operations

VECTOR PRODUCT first mention of the CROSS PRODUCT is

found on p. 61 of Vector Analysis, founded upon the lectures of J. Willard Gibbs, second edition, by Edwin Bidwell Wilson (1879-1964), published by Charles Scribner's Sons in 1909

the skew product is denoted by a cross as the direct product was by a dot

it is written: C = A X B and read A cross b for this reason it is often called the cross

product

THE CROSS PRODUCT a type of “multiplication” law that turns our vector space into a vector algebra A and B must be 3D vectors The result is a 3D vector with Length: |A × B|=|A||B|sinθ, where θ, is the angle between A and B and Orientation: A

× B is perpendicular to both A and B The choice of orientations is

made by the right hand rule.

Right hand Rule for the Direction of Cross Product Draw an arc starting

from the vector A and finishing on vector B

Curl your fingers the same way as the arc

Your right thumb points in the direction of the cross product

CCW rotation is in the +z direction

CW rotation is in the –z direction

HISTORY OF THE TORQUE The principle of moments is derived from Archimedes'

discovery of the operating principle of the lever He used to say, "Give me a place to stand and with a

lever I will move the whole world." In the lever one applies a force, in his day most often

human muscle, to an arm, a beam of some sort Archimedes noted that the amount of force applied to

the object, the moment of force, is defined as M = rF, where F is the applied force, and r is the distance from the applied force to object

The Law of the Lever According to ARCHIMEDES….. “Magnitudes are in equilibrium at distances reciprocally

proportional to their weights”. This is the statement of the Law of the Lever that

Archimedes gives in Propositions 6 and 7 of Book I of his work entitled On the Equilibrium of Planes.

Archimedes demonstrated mathematically that the ratio of the effort applied to the load raised is equal to the inverse ratio of the distances of the effort and load from the pivot or fulcrum of the lever

The force applied to a lever, multiplied by its distance from the lever's fulcrum, is the torque.

Newton’s Second Law of Rotation

the Second Law explains how a force acts on an object in linear motion

an object accelerates in the direction the force is moving it (F = ma)

Second Law of Rotation explains the relationship between the net external torque and the angular acceleration of an object

Rotational and Linear Example

What is a TORQUE Today?

TORQUE: FORCE APPLIED X WRENCH?

The lever arm is defined as the perpendicular distance from the axis of rotation to the line of action of the force.

RIGHT HAND RULE for TORQUE

WHY THE WRENCH?

TEACHING STRATEGIES

Matrices Physics Geometrical Representation Component Form Can we co-exist?