How to determine if two lines are equivalent using their parametric equations?

[DrunkApple]

Homework Statement

Show that r1(t) and r2(t) define the same line, where
$r_{1}$(t) = <3,-1,4> + t<8,12,-6>
$r_{2}$(t) = <11,11,-2> + t<4,6,-3>

Homework Equations

The Attempt at a Solution

I set $r_{1}$(t) = $r_{2}$(t) and got the value of t which is 2.
then I plugged that t value into $r_{1}$(t) and $r_{2}$(t) which both of them came out to be <19,23,-8>. Is this how to do it?

 

[SammyS]

Homework Statement

Show that r1(t) and r2(t) define the same line, where
$r_{1}$(t) = <3,-1,4> + t<8,12,-6>
$r_{2}$(t) = <11,11,-2> + t<4,6,-3>

Homework Equations

The Attempt at a Solution

I set $r_{1}$(t) = $r_{2}$(t) and got the value of t which is 2.
then I plugged that t value into $r_{1}$(t) and $r_{2}$(t) which both of them came out to be <19,23,-8>. Is this how to do it?

In general, you should use a different variable for the parameter in the two expressions; i.e.

r₁(t)=3,-1,4> + t<8,12,-6>

r₂(s)=<11,11,-2> + s<4,6,-3>​

All that you have shown is that the two lines intersect at <19,23,-8> .

See if you can find a linear relationship between s & t that makes the two lines equivalent.

 

[HallsofIvy]

$r_{1}$(t) = <3,-1,4> + t<8,12,-6>
$r_{2}$(t) = <11,11,-2> + t<4,6,-3>

As SammyS suggested, use another letter, say, s (in honor of SammyS, of course!) as parameter for the second equation. Then, where the lines intersect, we must have
x= 3+ 8t= 11+ 4s
y= -1+ 12t= 11+ 6s
z= 4- 6t= -2- 3t.

You can solve the first equation for, say, s as a function of t. Replace s in the other two equations with that and see what happens.