Changes

We are familiar with the basic geometry of the plane already, but a brief review will help illustrate the concepts we encounter in the next dimension.

We are familiar with the basic geometry of the plane already, but a brief review will help illustrate the concepts we encounter in the next dimension.

Any line in the xy plane can be described as <math>y=ax+b \ </math>, where <math>a \ </math> is the "slope" and <math>b \ </math> is the y-intercept.  When we raise the number of dimensions, we see how similar the equation of a plane is in three-space: <math>z=ax+by+c \ </math>, where <math>a \ </math> is the slope along the x-axis, etc.  See illustration.

Any line in the xy plane can be described as <math>y=ax+b \ </math>, where <math>a \ </math> is the "slope" and <math>b \ </math> is the y-intercept.  When we raise the number of dimensions, we see how similar the equation of a plane is in three-space: <math>z=ax+by+c \ </math>, where <math>a \ </math> is the slope along the x-axis, etc.   

Describing a line in three dimensions is trickier, and requires what is called a '''parametric equation'''.  This method of describing lines relies on some variable which isn't a coordinate, like time, to describe a point as it travels through space.  For example, if a point is at some location <math>(x_0, y_0, z_0) \ </math> at time <math>t=0 \ </math>, and then over 1 second travels <math>a \ </math> feet in the x direction, <math>b \ </math> feet in the y direction, and <math>c \ </math> feet in the z direction, we can descrive its location at any time <math>t \ </math> as <math>(x_0+at, y_0+bt, z_0+ct) \ </math>.  As it travels, this point traces out a line.

Describing a line in three dimensions is trickier, and requires what is called a '''parametric equation'''.  This method of describing lines relies on some variable which isn't a coordinate, like time, to describe a point as it travels through space.  For example, if a point is at some location <math>(x_0, y_0, z_0) \ </math> at time <math>t=0 \ </math>, and then over 1 second travels <math>a \ </math> feet in the x direction, <math>b \ </math> feet in the y direction, and <math>c \ </math> feet in the z direction, we can descrive its location at any time <math>t \ </math> as <math>(x_0+at, y_0+bt, z_0+ct) \ </math>.  As it travels, this point traces out a line.