Parametrization of a line examples

Example 1

Find a parametrization of the line through the points (3, 1, 2) and (1, 0, 5).

Solution: The line is parallel to the vector v = (3, 1, 2) - (1, 0, 5) = (2, 1,-3). Hence, a parametrization for the line is

x = (1, 0, 5) + t(2, 1,-3) for -∞ < t < ∞.
We could also write this as
x = (1 + 2t,t, 5 - 3t) for -∞ < t < ∞.
Or, if we write x = (x,y,z), we could write the parametric equation in component form as
x = 1 + 2t,
y = t,
z = 5 - 3t,
for -∞ < t < ∞.

Example 2

Find a parametrization for the line segment between the points (3, 1, 2) and (1, 0, 5).

Solution: The only difference from example 1 is that we need to restrict the range of t so that the line segment starts and ends at the given points. We can parametrize the line segment by

x = (1, 0, 5) + t(2, 1,-3) for 0 ≤ t ≤ 1.
(Or we could use any of the other forms of the parametric equation in example 1, as long as we restrict t to lie in the interval [0, 1].)