How would I find a vector parametric equation through these points?

[SMA_01]

Homework Statement

Find a vector parametric equation r(t) for the line through the points P=(3,0,4) and Q=(1,−3,9) for each of the given conditions on the parameter t.

I'm stuck on this one:

r(5)=P and r(8)=Q

Homework Equations

The Attempt at a Solution

I tried finding the parametric equations but it didn't work.

I don't really understand how to go about solving this. What is it asking exactly and how would I start?

Any help is appreciated.

 

[LCKurtz]

Can you find any parametric equation of the line through those two points? Let's start there; then we can talk.

 

[SMA_01]

@LCKurtz: Yeah I used the old method of finding the parametric equations, taking PQ as the vector, and using either P or Q as points. That didn't work out. I actually figured out how to do it, I just want to understand it. Like geometrically, what is the question saying?

 

[LCKurtz]

@LCKurtz: Yeah I used the old method of finding the parametric equations, taking PQ as the vector, and using either P or Q as points. That didn't work out. I actually figured out how to do it, I just want to understand it. Like geometrically, what is the question saying?

The "standard" parametric equation with that direction vector and starting at P is at the point P when t = 0 and Q when t = 1. The point of the problem is to rescale the parameter t so that it is at point P when t = 3 and Q when t = 8.

 

[HallsofIvy]

Suppose you just start with the general linear formulas x= at+ b, y= ct+ d, z= et+ f and use the facts that
x(5)= 3, y(5)= 0, z(5)= 4 and x(8)= 1, y(8)= −3, z(8)= 9.

You have three sets of two equations in two unknowns.

 

[SMA_01]

@HallsofIvy and LCKurtz: Thanks, so basically I'm re-scaling the original linear equations to fit into the new parameters? Sorry, I'm just trying to understand the geometrical aspect.

 

[HallsofIvy]

Well, a translation, to put "0" at "5", and then a scaling, to stretch "0 to 1" to "5 to 8".

 

[LCKurtz]

@HallsofIvy and LCKurtz: Thanks, so basically I'm re-scaling the original linear equations to fit into the new parameters? Sorry, I'm just trying to understand the geometrical aspect.

Think about what happens if you take the "standard" equation starting at P and use PQ as the direction vector, but then replace t by (t-5)/3.