Proving derivatives of a parametrized line are parallel

In summary, if σ(t) is a parametrization of a line, then σ''(t) is parallel to σ'(t). This can be shown by expressing σ(t) as vt + a or as f(t)D + a, where v and D are constant vectors, and a is a constant position vector. In both cases, σ'(t) and σ''(t) are parallel.
  • #1
V0ODO0CH1LD
278
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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?
 
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  • #2
V0ODO0CH1LD said:

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?
 
  • #3
Ah, thank you! But both ##\vec{D}## and ##\vec{a}## are still just regular vectors, right?
 
  • #4
V0ODO0CH1LD said:
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.
 

1. What is a parametrized line?

A parametrized line is a line that is described by a set of equations using a parameter, usually denoted as t. This allows us to represent points on the line as a function of the parameter t, making it easier to calculate various properties of the line.

2. How do you prove that two parametrized lines are parallel?

To prove that two parametrized lines are parallel, we need to show that their direction vectors are parallel. This can be done by calculating the derivatives of the parametric equations for each line and showing that they are scalar multiples of each other.

3. Can parametrized lines ever be perpendicular?

No, parametrized lines cannot be perpendicular because their direction vectors must be parallel in order for them to be considered parallel. Perpendicular lines have direction vectors that are perpendicular to each other.

4. Why is it important to prove that derivatives of parametrized lines are parallel?

Proving that the derivatives of parametrized lines are parallel can help us solve problems involving these lines, such as finding the point of intersection or the angle between them. It also allows us to extend our understanding of derivatives to a wider range of functions.

5. Are there any other methods for proving that parametrized lines are parallel?

Yes, there are other methods for proving that parametrized lines are parallel, such as using the dot product or cross product of their direction vectors. However, using derivatives is a commonly used and straightforward method for this type of proof.

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