Line through a Point parallel to a Vector:

Let r₀ represent the point P and r be a vector to a general point, also on the line L.

The vector equation of L is r = r₀ + t*a = 2i - 3j + k + t*[5i + j - 2k].

In parametric form, we have

x = x₀ + t*a = 2 + 5t

y = y₀ + t*b= -3 + t

z = z₀ + t*c = 1 - 2t