Representing A Vector in The Standard Basis
A point in space in a Cartesian coordinate system may also be represented by a position vector, which can be thought of as an arrow pointing from the origin of the coordinate system to the point. If the coordinates represent spatial positions (displacements), it is common to represent the vector from the origin to the point of interest as . In two dimensions, the vector from the origin to the point with Cartesian coordinates (x, y) can be written as:
where, and are unit vectors in the direction of the x-axis and y-axis respectively, generally referred to as the standard basis (in some application areas these may also be referred to as versors). Similarly, in three dimensions, the vector from the origin to the point with Cartesian coordinates can be written as:
where is the unit vector in the direction of the z-axis.
There is no natural interpretation of multiplying vectors to obtain another vector that works in all dimensions, however there is a way to use complex numbers to provide such a multiplication. In a two dimensional cartesian plane, identify the point with coordinates (x, y) with the complex number z = x + iy. Here, i is the complex number whose square is the real number −1 and is identified with the point with coordinates (0,1), so it is not the unit vector in the direction of the x-axis (this confusion is just an unfortunate historical accident). Since the complex numbers can be multiplied giving another complex number, this identification provides a means to "multiply" vectors. In a three dimensional cartesian space a similar identification can be made with a subset of the quaternions.
Read more about this topic: Cartesian Coordinate System
Famous quotes containing the words representing a, representing, standard and/or basis:
“He who has learned what is commonly considered the whole art of painting, that is, the art of representing any natural object faithfully, has as yet only learned the language by which his thoughts are to be expressed.”
—John Ruskin (1819–1900)
“Brave people may be persuaded to an action by representing it as being more dangerous than it really is.”
—Friedrich Nietzsche (1844–1900)
“[The Declaration of Independence] meant to set up a standard maxim for free society, which should be familiar to all, and revered by all; constantly looked to, constantly labored for, and even though never perfectly attained, constantly approximated, and thereby constantly spreading and deepening its influence, and augmenting the happiness and value of life to all people of all colors everywhere.”
—Abraham Lincoln (1809–1865)
“The basis of world peace is the teaching which runs through almost all the great religions of the world. “Love thy neighbor as thyself.” Christ, some of the other great Jewish teachers, Buddha, all preached it. Their followers forgot it. What is the trouble between capital and labor, what is the trouble in many of our communities, but rather a universal forgetting that this teaching is one of our first obligations.”
—Eleanor Roosevelt (1884–1962)