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

1 Answer
Aug 20, 2014

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^3R3, call it P = (x_0,y_0,z_0)P=(x0,y0,z0).

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 + tux=x0+tu
y = y_0 + tvy=y0+tv (-oo < t < oo)(<t<)
z = z_0 + twz=z0+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 + tux=2+tu
y = 4 + 3vy=4+3v (-oo < t < oo)(<t<)
z = 6 + 5wz=6+5w

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