Equations of Lines in Space

These are pictures/animations to visualize the vector, parametric and symmetric equations of a line. All of them are for the same line through the point (2, 5, 1) with direction vector <3, 2, 4>. There are two views of each to get a better feeling of 3D.

The first pictures represent the vector equation r=<2, 5, 1> +t<3, 2, 4>. The initial position vector r₀=<2, 5, 1> is green and the position vector r=<x,y,z> tracing  the line is blue. The animation is from t=0 to t=1.

The second set of pictures represents the parametric equations. The point (2, 5, 1) is green and the arbitrary point (x, y, z) tracing the line is blue. The animation is again from t=0 to t=1.

The third set of pictures represents the symmetric equations. They describe the line (red) as the intersection of the three planes given by (x-2)/3=(y-5)/2 (blue), (x-2)/3=(z-1)/4 (yellow) and (z-1)/4=(y-5)/2 (green).Note that two intersecting planes would suffice to describe a line. Here, nothing is moving as we eliminated the parameter t.