Parametric Equation of a line, Conditions

[Destroxia]

Homework Statement

Find parametric equations for the line through P1 and P2 and also for the line segment joining those points. Also find the conditions for the parametric equations.

a) P1(0, 2) and P2(-4,-4)

b) P1(7,-3,9) and P2(7, -3, 1)

Homework Equations

Equation of a line (Vector Form): $\vec r = \vec r_0 + t\vec v$[/B]

The Attempt at a Solution

[/B]
I've already solved the problem, my issue just lies in finding the conditions for the parametric equations as well.

a) x = -4t, y = 2 - 6t

b) x = 7, y = -3, z = 9 - 8t

Wouldn't the parametric equation work from $-\infty ≤ t ≤ \infty$

I see no reason why some parallel direction vector wouldn't be able to be multiplied by some scalar to equal the line vector. Any multiple, negative or positive should be able to give me any line that passes through those points.

So am I correct in my assumption that the conditions should be from $-\infty ≤ t ≤ \infty$

 

[Mark44]

Homework Statement

Find parametric equations for the line through P1 and P2 and also for the line segment joining those points. Also find the conditions for the parametric equations.

a) P1(0, 2) and P2(-4,-4)

b) P1(7,-3,9) and P2(7, -3, 1)

Homework Equations

Equation of a line (Vector Form): $\vec r = \vec r_0 + t\vec v$[/B]

The Attempt at a Solution

[/B]
I've already solved the problem, my issue just lies in finding the conditions for the parametric equations as well.

a) x = -4t, y = 2 - 6t

b) x = 7, y = -3, z = 9 - 8t

Wouldn't the parametric equation work from $-\infty ≤ t ≤ \infty$

I see no reason why some parallel direction vector wouldn't be able to be multiplied by some scalar to equal the line vector. Any multiple, negative or positive should be able to give me any line that passes through those points.

So am I correct in my assumption that the conditions should be from $-\infty ≤ t ≤ \infty$

For the lines through those two sets of points, yes, but the conditions they're asking about are for the line segments. For the first problem, if t = 0, you get point P₂. What value of t (the parameter) gets you P₁? All of the t values in between get you the rest of the points on that line segment.

 

[Destroxia]

For the lines through those two sets of points, yes, but the conditions they're asking about are for the line segments. For the first problem, if t = 0, you get point P₂. What value of t (the parameter) gets you P₁? All of the t values in between get you the rest of the points on that line segment.

I think I see what you mean... so we are just solving for t at each end of the line segment?

 

[MidgetDwarf]

it is [0,1] for all t, any linear algebra book will have this fact in the first section.