Step By Step Calculus » 2.6 - The Rectangular Coordinate System

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Synopsis
  • A plane refers to any flat (two dimensional) surface.
  • A (two dimensional) co-ordinate system is a method of assigning location to point on a plane.
  • The rectangular co-ordinate system in the plane is also called the Cartesian co-ordinate system in the plane.
  • This co-ordinate system is defined by a point called the origin and two perpendicular lines crossing at the origin O O, called the x x-axis and the y y-axis.
  • The location of a point P P on the plane is defined by a unique pair of numbers (a,b) (a,b), where a a, b b are the signed (or directed) distances of P P from the y y- and x x-axis respectively. We often relabel this point P(a,b) P(a,b) or just (a,b) (a,b) when the location of this point is to be emphasized.
  • The origin O O is the unique point located at \left(0,0\right) \left(0,0\right).
  • The set of possible locations in the rectangular co-ordinate system is \mathbb{R}^{2} \mathbb{R}^{2} since each location is equated with two numbers, the former giving the distance along the x x-axis and the later giving the distance along the y y-axis.
  • The distance d d between two points P_0(x_0,y_0) P_0(x_0,y_0) and P_1(x_1,y_1) P_1(x_1,y_1) in the plane is
    \displaystyle \textbf{(Distance Formula)} \qquad d(P_0,P_1)=| \, P_0\, P_1 \, |=\sqrt{(x_1-x_0)^2+(y_1-y_0)^2} \, .
    \displaystyle \textbf{(Distance Formula)} \qquad d(P_0,P_1)=| \, P_0\, P_1 \, |=\sqrt{(x_1-x_0)^2+(y_1-y_0)^2} \, .
  • For example, the set of points in the plane which lie on or within a circle centered at the origin with radius 33 is given by the set of locations \{P:|PO|\leq 3\}= \{(x,y)\;|\; x^2+y^2\leq 9,\;\;x,y\in\mathbb{R} \} \{P:|PO|\leq 3\}= \{(x,y)\;|\; x^2+y^2\leq 9,\;\;x,y\in\mathbb{R} \} .
  • The midpoint between the points P_0(x_0,y_0) P_0(x_0,y_0) and P_1(x_1,y_1) P_1(x_1,y_1) is
    \displaystyle \textbf{(Midpoint Formula)} \qquad P_{\frac12}\left(x_{\frac12}, y_{\frac12}\right)=\left(\frac{x_0+x_1}{2},\frac{y_0+y_1}{2}\right).
    \displaystyle \textbf{(Midpoint Formula)} \qquad P_{\frac12}\left(x_{\frac12}, y_{\frac12}\right)=\left(\frac{x_0+x_1}{2},\frac{y_0+y_1}{2}\right).
  • The point P_pP_p which lies at the fraction pp of the full distance from P_0(x_0,y_0)P_0(x_0,y_0) to point P_1(x_1,y_1)P_1(x_1,y_1) is
    \displaystyle \textbf{($p$-point Formula)} \qquad P_p=\left((1-p)x_0+px_1, (1-p)y_0+py_1\right).
    \displaystyle \textbf{($p$-point Formula)} \qquad P_p=\left((1-p)x_0+px_1, (1-p)y_0+py_1\right).
  • The point (x,0)(x,0) is the projection of the point (x,y)(x,y) on the xx-axis and the point (0,y)(0,y) is the projection of the point (x,y)(x,y) on the yy-axis.