Vector Video Tutorial

Vector Algebra

Vector Algebra

A scalar is a value that can be represented by a single number.
A vector is also quantity that has both magnitude (size) and direction. Vectors are often written in bold type, to distinguish them from scalars. Velocity of a moving point is an example of a vector quantity.In two dimensional space a vector may be represented by two scalar components, in three dimensions a vector may be represented by three scalar components. Most simply these are Cartesian coordinates.  However in 2D vectors can be written in polar coordinates and in 3D they can be written in spherical or cylindrical coordinates.

Elementary Vector Algebra Vector Arithmetic Notes
Vectors follow obvious rules of addition and subtraction. Also the multiplication of a vector by a scalar is straightforward. For example if a = (a₁,a₂,a₃),b = (b₁,b₂,b₃) are vectors and a is a scalar then

a + b = (a1,a2,a3)+(b1,b2,b3) = (a1+b1,a2+b2,a3+b3)

a - b = (a1,a2,a3)-(b1,b2,b3) = (a1-b1,a2-b2,a3-b3)

aa = (aa1,aa2,aa3)

Dot Product
The dot product of two vectors a and b is defined as follows:

a . b = (a1,a2,a3).(b1,b2,b3) = a1 b1+ a2 b2+ a3 b3

Cross Product
The cross product of two vectors a and b is defined as follows:

a ×b = (a1,a2,a3) ×(b1,b2,b3) = (a2 b3 - a3 b2, a3 b1 - a1 b3, a1 b2 -a2 b1)

The Gradient

The gradient is a vector operation which operates on a scalar function to produce a vector whose magnitude is the maximum rate of change of the function at the point of the gradient and which is pointed in the direction of that maximum rate of change. In rectangular coordinates the gradient of function f(x,y,z) is:

If S is a surface of constant value for the function f(x,y,z) then the gradient on the surface defines a vector which is normal to the surface.
http://hyperphysics.phy-astr.gsu.edu/hbase/gradi.html

Scientific Revolution

701- Jethro Tull invents the seed drill.

1709- Bartolomeo Cristofori invents the piano.

1711- Englishmen, John Shore invents the tuning fork.

1712- Thomas Newcomen patents the atmospheric steam engine.

1717- Edmond Halley invents the diving bell.

1722- French C. Hopffer patents the fire extinguisher.

1724- Gabriel Fahrenheit invents the first mercury thermometer.

1733- John Kay invents the flying shuttle.

1745- E.G. von Kleist invents the leyden jar, the first electrical capacitor.

The Scientific Revolution

The "Scientific Revolution" is a period of about 150 years during which the classical world view presented in the works of Aristotle, Ptolemy, and Galen was replaced by the fundamental ideas of modern science. Historians generally date it from the publication of Copernicus' De revolutionibusin 1543 to that of Newton's Principiain 1687. The most important new ideas developed during this time were:
- the mechanical philosophy
- the corpuscular philosophy, including atomism
- the experimental philosophy

Important persons:

Nicolaus Copernicus (1473-1543) Poland, astronomy
William Gilbert (1540-1603) England, magnetism
Tycho Brahe (1546-1601) Denmark, astronomy
Francis Bacon (1561-1627) England, method
Galileo Galilei (1564-1642) Italy, astronomy, mechanics
Johann Kepler (1571-1630) Germany, astronomy, optics
Pierre Gassendi (1592-1655) England, atomism
Rene Descartes (1596-1650) France, all fields
Robert Hooke (1635-1703) England, microscopy
Isaac Newton (1642-1727) England, physics, mathematics

Important events and publications:

1543 Copernicus, On the Revolution of the Heavenly Spheres
1572 Tycho Brahe observes a new star (nova)
1573 Brahe, On the new star
1577 Tycho Brahe measures the parallax of a comet
1580 Tycho Brahe begins construction of his observatory on Hven
1588 Brahe, On recent phenomena of the aetherial world
1596 Kepler, Mysterium cosmographicum
1600 Gilbert, On the magnet
1602 Brahe, Introduction to the New Astronomy
1605 Bacon, The Advancement of Learning
1609 Kepler, The New Astronomy
1610 Galileo, The Starry Messenger
1613 Galileo, Letters on Sunspots
1616 Copernicus' book is put on the Index of Prohibited Books
1619 Kepler, On the Harmony of the World
1620 Bacon, New Organon
1627 Bacon, New Atlantis
1632 Galileo, Dialogue concerning the Two Chief World Systems
1637 Descartes, Discourse on Method, Geometry, Optics
1638 Galileo, Discourse on Two New Sciences
1642 Descartes, Principles of Philosophy
1649 Gassendi, Synthesis of the Epicurean Philosophy
1660 Royal Society founded in London
1661 Robert Boyle, The Sceptical Chymist
1666 Academy of Sciences founded in Paris
1665-6 Newton's annus mirabilis
1687 Newton, Mathematical Principles of Natural Philosophy