# Chapter 14 - Calculus of Vector-Valued Functions - 14.1 Vector-Valued Functions - Exercises - Page 710: 16

See the explanation below.

#### Work Step by Step

The projection onto the xy-plane is traced by $\langle \sin t,0\rangle$ which is a segment $[-1,1]$ on the x-axis since $-1\leq \sin t \leq 1$. The projection onto the xz-plane is the circle itself. We put $$x=\sin t, z-4=\cos t$$ hence we get $$x^2+(z-4)^2=1$$ which is a circle of radius $1$ centered at $(0,0,4)$. The projection onto the yz-plane is traced by $\langle 0,0,4+\cos t\rangle$ which is a segment $[3,5]$ on the z-axis since $3=4-1\leq 4+ \cos t \leq 4+1=5$.

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