Final answer:
The parametric equations of the line that passes through the points (0, 1, 2) and (3, 1, -4) are x(t) = 3t, y(t) = 1, and z(t) = 2 - 6t, where t is any real number.
Step-by-step explanation:
To find the parametric equations for the line passing through the points (0, 1, 2) and (3, 1, -4), we first find the direction vector by subtracting the coordinates of the first point from the second, resulting in the direction vector (3, 0, -6). We can then express the parametric equations using the parameter t, where the line passing through the point (0, 1, 2) in the direction of vector (3, 0, -6) is given by x(t) = 0 + 3t, y(t) = 1 + 0t, and z(t) = 2 - 6t. Therefore, the parametric equations of the line are:
x(t) = 3t
y(t) = 1
z(t) = 2 - 6t
These equations imply that for any real number value of t, the point (3t, 1, 2-6t) lies on the line.