Which came first:
Vector Product or Torque?
by Antonia Katsinos
VECTOR ALGEBRA vs. TORQUE
CROSS 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 TRADITIONS
THE 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)
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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
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Matrices
Physics
Geometrical Representation
Component Form
Can we co-exist?