Lines

Pre-requisites: Vector Arithmetic

First of all, forget about y = m x + b. Forget about slope, y-intercepts, and all that stuff. Not only do they not work for lines in R3, they don't even work for all lines in the plane (for example, x = 3).

The right way to think about lines is as straight curves . A (parameterized) curve is just a function which tells you where (what point) the curve is at at every time (parameter) value. For example, sigma(t) = ( t, t2, cos t ) is the curve describing a particle which is at position (0,0,1) at time 0, (1,1,cos 1) at time 1, etc.

A line in R3 is a curve of the form l(t) = v t + p = ( v1 t + p1, v2 t + p2, v2 t + p2 ) v, or any scalar multiple of v, is called a vector along the line l. To see why, consider any two points along the line: call them l(t1) and l(t2). Then the vector between them will equal v t1 + p - v t2 - p = v ( t1 - t2 ), which is a multiple of v.

To find the equation of some line, all you have to do is find some point on it and call it p and then find some vector along the line and call it v, and then plug it all into the formula l(t) = v t + p. For example, the line between the two points P and Q is lPQ(t) = (Q - P) t + P. It is obvious from this description that there are many equations that describe the same line; this is prefectly fine and will in fact always be the case when an object is described parameterically.

Very rarely will you want to use the implicit definition of a line. For one thing, it is unwieldly: the implicit definition of a line consists of two equations. For example, the z-axis is described implicitly by the two equations x=0 and y=0. In general, the implicit description of a line will look like

A1 x + B1 y + C1 z = D1 A2 x + B2 y + C2 z = D2
One way to think about this is to say "Well, the parametric equation for a line is a vector valued equation, so when I break it up I get three equations in the four variables x,y,z (the components of l(t)) and t. If I eliminate t, I'll get two equations in the three variables x,y, and z." Another way to think about it is as the intersection of two planes.

Exercises:

  1. Find the line running along the y-axis.
  2. Find the line between the point (1,2,3) and the point (4,5,6).
  3. Find a vector lying on the line l(t) = (1 - t, 1 + t, 3- 7t).
  4. Show that if a line contains the points P and Q, then it contains the point (P+Q)/2.
  5. Convert the line l(t) = (5,4,2)t + (1,0,-3) to an implicit description.
  6. Convert the line x + y + z = 3, x - y = 7 to a parametric description.
  7. Convert the line x + 2 = y/5 + 3 = z/7 - 10 to a parametric description.
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thomasc@athena.mit.edu
last modified 1 July 1997