Answer to Question #70409 in Analytic Geometry for sajid

Question

Prove that If the tangent vector of the parameterized curve is constant, the image of the curve is (part of) a straight line.

Answer 1

Parameterized curve: a(t)=<x(t),y(t),z(t)>.
Tangent vector: a^' (t)=<x^' (t),y^' (t),z^' (t)>.
If a^' (t)=<x_0,y_0,z_0> (where x_0,y_0,z_0 are constants) then,
a(t)=∫〖a^' (t)dt〗=<x_0 t+x_1,y_0 t+y_1,z_0 t+z_1>.
If <x_0,y_0,z_0>≠0, this is the parametric equation of the straight line parallel to <x_0,y_0,z_0> and passing through the point with position vector
<x_1,y_1,z_1>.
If <x_0,y_0,z_0>=0, the image of the curve is the single point <x_1,y_1,z_1>.