What is meant by the parametric equations of a line in three-dimensional space?

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Answer 1 · 2014-08-20T19:28:30

We want to find the parametric equations of the line L passing through the point P and parallel to a vector A.

Let us take a 3-dimensional point in $R^{3}$ $R^3$ , call it $P = (x_{0}, y_{0}, z_{0})$ $P = (x_0,y_0,z_0)$ .

A line L is drawn such that it passes through P and is parallel to the vector $A = (u, v, w)$ $A = (u,v,w)$ .

3 parametric equations can be written which express the components:

$x = x_{0} + t u$ $x = x_0 + tu$
$y = y_{0} + t v$ $y = y_0 + tv$ $(- \infty < t < \infty)$ $(-oo < t < oo)$
$z = z_{0} + t w$ $z = z_0 + tw$

As an example, with a point $P = (2, 4, 6)$ $P = (2,4,6)$ and vector $A = (1, 3, 5)$ $A = (1,3,5)$ , we have the following parametric equations:

$x = 2 + t u$ $x = 2 + tu$
$y = 4 + 3 v$ $y = 4 + 3v$ $(- \infty < t < \infty)$ $(-oo < t < oo)$
$z = 6 + 5 w$ $z = 6 + 5w$