Describe the curve defined by the vector function

r$($t$)=$

$1+$ t$,5+3$t$,5+3$t

$.$

Solution

The corresponding parametric equations are

x $=$

$,$ y $=5+3$t$,$ z $=$

$,$

which we recognize from the equations

x $=$ x$0+$ at$,$ y $=$ y$0+$ bt$,$ z $=$ z$0+$ ct

as parametric equations of a line passing through the point

$(1,5,5)$

and parallel to the vector

$1,3,3$

$.$

Alternatively, we could observe that the function can be written as

r $=$ r$0+$ tv$,$

where

r$0=$

$1,5,5$

and

v $=$

$,$

and this is the vector equation of a line as given by the equation

r $=$ r$0+$ tv$.$