Proving derivatives of a parametrized line are parallel

[V0ODO0CH1LD]

Homework Statement

Show that if σ(t) for (t in I) is a parametrization of a line, then σ''(t) is parallel to σ'(t).

Homework Equations

The Attempt at a Solution

I thought that if σ(t) is a parametrization of a line then it could be expressed as σ(t) = vt + a, but then σ'(t) = v and σ''(t) = 0. Can two vectors be parallel? Is it because the distance between them is constant? Or did I make a mistake earlier on?

 

[LCKurtz]

Homework Statement

Show that if σ(t) for (t in I) is a parametrization of a line, then σ''(t) is parallel to σ'(t).

Homework Equations

The Attempt at a Solution

I thought that if σ(t) is a parametrization of a line then it could be expressed as σ(t) = vt + a, but then σ'(t) = v and σ''(t) = 0. Can two vectors be parallel? Is it because the distance between them is constant? Or did I make a mistake earlier on?

That would be the equation of a line where the point $\sigma(t)$ moves with constant velocity. But what about something of the form $\sigma(t) = f(t)\vec D + \vec a$? Wouldn't that give a straight line too?

 

[V0ODO0CH1LD]

Ah, thank you! But both $\vec{D}$ and $\vec{a}$ are still just regular vectors, right?

 

[LCKurtz]

Ah, thank you! But both $\vec{D}$ and $\vec{a}$ are still just regular vectors, right?

Yes. $\vec D$ is a direction vector and $\vec a$ is a position vector to a point on the line. Both vectors are constants.