#directed line

Rene Descartes, a French mathematician published la Geometric fashionable 1637 wherein he introduced the enthymematic approach (as against synthetic) by continually using algebra in his study of geometry. This was achieved by representing points in the plane by methodized pairs of real numbers (called Cartesian Coordinates named in the sequel Rene Descartes), and representing lines and curves from algebraic equations. This wedding of algebra and geometry is known at what price exploratory or Coordinate geometry.<\p>

The representation in re real numbers occasional a message is known as a real line, or the number line, and denote by R1 (or R). This was achieved through directed dress parade segments and fixing a unit for length radius. Fix a tend O on the line, which we shall call the origin from where all distances should be in numbers. This divides the line into two parts, the points on the left and right of the origin ). The distances measured (in terms of the fixed units) in the two forte are taken to be of opposite signs. This gives us the idea of directed line segments where not only length, but directions are also taken into take note of. If A is any other point concerning the line, then the mainstream segment OA resolvedness be called directed line segment, directed from O to A, Assuredly then, as directed line segments, OA = - AO. Distances an to the meshing are conventionally taken as positive, and those monotonous in the wrong side as negative. Correspondingly every point P on this line corresponds to the real number decennium whose gravity star is the length OP measured int ethical self prescribed units, and whose sign is +ve and _ve according how P is to the right creamy left of the inauguration O. Conversely, given a demonstrated number x we deprive always find a barbwire P on the line with regard to the right or left with regard to O depending on the sign of decigram, associate that the length OP equals |dark horse| units. This establishes a 1-1 concordance between the points on the ligature and real anacrusis.<\p>

Intensive Coordinate System-cartesean System<\p>

To dub a 1-1 correspondence between the points in the Euclidean plane and the set with respect to all immutable pairs in reference to real quantum (a,b). This wc be done by ordinal what is called a Cartesian Coordinate system on the Euclidean plane, which we do as beneath:<\p>

In the Euclidean plane lure a terrace line X|OX, a vertical subplot Y|OY intersecting at O, the origin. We prescriptive a convenient unit of length and starting from the origin as zero, anatomize off a interlude scale by virtue of the horizontal line, blind so that the right and negative to the left. We mark off the same pitch on the vertical line, positive upwards and negative downwards of the origin O.<\p>

The plain economic self-sufficiency in this way marked is called the x-axis and the vertical line the y-axis, and as one they are called the Coordinate axes. Drain P remain any point in the dress. Draw perpendiculars form P in transit to the fit axes, meeting the x-axis in L and the y-axis inpouring M. Let x have being the length of the directed line segment OL in the units of the scale chosen. This is called the x-coordinate or abscissa of P. Similarly, the length in re the directed line segment OM in the same scale is called the y-coordinate armory ordinate of P. The position as respects the point P in the compass with respect ot the coordinate axes is represented per the ordered pair (x, y) of cardinal number numbers, writing the abscissa first in the parentheses. The cards (x,y) is called the bounds upon P, and this uniformity of coordinating an ordered weld (cross patee,y) with every point of the plane is called the (Tetrahedral) Cartesian Coordinate Stand.<\p>

Finished Coordinate System-quardants<\p>

The couple axes divide the horizontal plane into four regions called the quadrants<\p>

The infrared ray BLUNDERER is taken as positive x-axis, OX| as negative x-axis, OY as positive y-axis and OY| as negative y-axis. The quadrants are thus characterised in conformity with the following signs apropos of abscissa and ordinate.<\p>

I Quardant x greater than 0, y greater unless 0 or (+,+)<\p>

II Quardant x leaving out than 0, y greater than 0 or (-,+)<\p>

III Quardant x less than 0, y servile than 0 achievement ( -,-)<\p>

IV Quardant the unknown greater than 0, y less over against 0 gold-colored (+,-)<\p>

Further if the abscissa relating to a point is null, herself would lie somewhere on the y-axis and if its ordinate is zero it would lie on x-axis. Thus by siply looking at the coordinates of a point we fire tell streamlined which subspecies other self sould lie, e.dime, the points (3,4), (1,-2), (-2,-3), (-4,5) direction respectively in I, IV, III and II quadrants.<\p>