Answer
$r(t) =\lt 4 \cos t, 4 \sin t, 5-4 \cos t \gt$
Work Step by Step
The parametric equation for $x=4 \cos t; y=4 \sin t$ defines the first equation for $x^2+y^2=16$ ...(1) ; the equation of a circle having radius $4$.
Now, let us consider the second equation; that is, $x+z=5$ ...(2)
After solving these above equations, we get $z=5 -x$ and $z=5-4 \cos t$
Hence, we get the parametric equations in vector form as follows:
$r(t) =\lt 4 \cos t, 4 \sin t, 5-4 \cos t \gt$
or, $r(t) =4 \cos t i+ 4 \sin tj+(5-4 \cos t) k$
