If two objects travel through space along two different curves, it is often important to know whether they will collide. $($Will a missile hit its moving target? Will two aircraft collide?$)$ Their paths might intersect, but we need to know whether the objects are in the same position at the same time.

Suppose two particles travel along the following space curves.

r$1($t$)=$

t$,$ t$2,$ t$3$

$,$ r$2($t$)=$

$1+4$t$,1+16$t$,1+52$t

$$

$$for

t $0$

Find the points at which their paths intersect. $($If an answer does not exist, enter DNE.$)$

smaller x$-$value

$($x$,$ y$,$ z$)$

$=$

$1,1,1$

larger x$-$value

$($x$,$ y$,$ z$)$

$=$

$3,9,27$

Find the time$($s$)$ when the particles collide. $($Enter your answers as a comma$-$separated list. If an answer does not exist, enter DNE.$)$

t $=$

$3,1$