# eq. of line in space.

• Dec 29th 2008, 10:58 PM
tombrownington
eq. of line in space.
Give two points A and B in 3d space with co-ordinates:
A(a,b,c) and B(a',b',c'), what is the equation of the line passing through both points?

Thanks for the help.
• Dec 30th 2008, 02:37 AM
james_bond
See this maybe it helps.

The parametric equations for a line passing through $\displaystyle (x_1, y_1, z_1)$, $\displaystyle (x_2, y_2, z_2)$ are:
$\displaystyle \begin{cases}x = x_1 + (x_2 - x_1)\cdot t \\ y = y_1 + (y_2 - y_1)\cdot t\\ z = z_1 + (z_2 - z_1)\cdot t \end{cases}$
• Dec 30th 2008, 02:49 AM
HallsofIvy
Quote:

Originally Posted by james_bond
See this maybe it helps.

The parametric equations for a line passing through $\displaystyle (x_1, y_1, z_1)$, $\displaystyle (x_2, y_2, z_2)$ are:
$\displaystyle \begin{cases}x = x_1 + (x_2 - x_1)\cdot t \\ y = y_1 + (y_2 - y_1)\cdot t\\ z = z_1 + (z_2 - z_1)\cdot t \end{cases}$

You can also write a line in three dimensions in "symmetric" form.
Solving each of those equations, for t,

$\displaystyle \frac{x- x_1}{x_2- x_1}= \frac{y- y_1}{y_2- y_1}= \frac{z- z_1}{z_2- z_1}$

I notice you asked for "the" equation of a line in three dimensions. Because a line is only one dimension, you need to "reduce" two dimensions and so need either two equations (3- 2= 1), as I give or introduce a new parameter, t, giving 4 variables with 3 equations: 4- 3= 1.