# 1. Coordinate Systems

## a. Rectangular Coordinates - 2D, 3D and nD

## 2. 3D Rectangular Coordinates

The Rectangular Coordinate System in space adds a third dimension, \(z\), to the two directions, \(x\) and \(y\), in the two-dimensional Rectangular Coordinate System. It represents the three dimensions of the natural world. There are now three perpendicular axes, the \(x\)-axis and the \(y\)-axis are horizontal while the \(z\)-axis is vertical. Points are specified by giving an ordered triple \((x,y,z)\). The \(x\)-coordinate is positive in front of the origin and negative behind. The \(y\)-coordinate is positive to the right of the origin and negative to the left. The \(z\)-coordinate is positive above the origin and negative below. This relation between the positive axes is called a right handed coordinate system because when you hold the fingers of your right hand along the \(x\)-axis and rotate your fingers toward the \(y\)-axis, your thumb points along the \(z\)-axis. The \(x\)- and \(y\)-axes span the \(xy\)-plane. The \(x\)- and \(z\)-axes span the \(xz\)-plane. The \(y\)- and \(z\)-axes span the \(yz\)-plane. Since it takes \(3\) \(\mathbb{R}\)eal numbers to specify a point, space is also called \(\mathbb{R}^3\). In the plot at the right, we have marked in the point \(P=(4,3,2)\). Rotate the plot with your mouse.

The intersecting coordinate planes create eight octants, labeled from I, through VIII. There is no standard ordering of the octants except that octant I is the octant where \(x\), \(y\) and \(z\) are all positive. However, here is the choice that will be used here:

I | II | III | IV | V | VI | VII | VIII | |
---|---|---|---|---|---|---|---|---|

\(x\) | \(+\) | \(-\) | \(-\) | \(+\) | \(+\) | \(-\) | \(-\) | \(+\) |

\(y\) | \(+\) | \(+\) | \(-\) | \(-\) | \(+\) | \(+\) | \(-\) | \(-\) |

\(z\) | \(+\) | \(+\) | \(+\) | \(+\) | \(-\) | \(-\) | \(-\) | \(-\) |

When working with symbols, the coordinates of a point are frequently written using subscripts. Thus a point \(P\) may be written as \(P=(p_1, p_2, p_3)\).

Identify the coordinates of each point in the following plot:

Rotate the plot with your mouse.

\(P=\) | \((1,2,3)\) | \((2,1,3)\) | \((3,1,2)\) | \((3,2,1)\) |
---|---|---|---|---|

\(Q=\) | \((-2,3,3)\) | \((-2,-3,3)\) | \((-3,-2,-3)\) | \((2,-3,3)\) |

\(R=\) | \((4,-2,-2)\) | \((-4,-2,-2)\) | \((4,-2,2)\) | \((-4,-2,2)\) |

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