Describe the curve defined by the vector function r(t) = 3 + t, 5 + 4t, −2 + 3t . solution the corresponding parametric equations are x = 3+t , y = 5 + 4t, z = −2+3t which we recognize from these equations as parametric equations of a line passing through the point (3, 5, −2) and parallel to the vector 1, 4, 3 . alternatively, we could observe that the function can be written as r = r0 + tv, where r0 = 3, 5, −2 and v = t⟨1, 4, 3⟩ , and this is the vector equation of a line as given by this equation.