Where does the ball land and with what speed? A ball is thrown eastward into the air from the origin (in the direction of the positive x-axis). The initial velocity = $<50, 0, 80>$ Acceleration = $<0, -4, -32>$ Would it make sense to find the time that the ball lands by setting the z-component of the velocity vector equal to zero. That is, $-32t + 80 =0$, then $t = \frac{5}{2}$ and then plug it into the corresponding vectors?

No, we presume the ground is horizontal, so it lands when $x=0$. You need to find $z(t)$. There will be two roots to $z(t)=0$. One will be $t=0$ as it started at zero elevation. The other is the one you want. If you write the equation, $z$ will be quadratic in time, so you use the quadratic formula to find the root.