Answer
Space curve (a) matches Figure 8 (C)
Space curve (b) matches Figure 8 (B)
Space curve (c) matches Figure 8 (A)
Work Step by Step
(a) ${{\bf{r}}_1}\left( t \right) = \left( {\cos 2t,\cos t,\sin t} \right)$.
From the result in Exercise 11 and examining the curve and its projection, we conclude that the space curve is Figure 8 (C).
(b) ${{\bf{r}}_2}\left( t \right) = \left( {t,\cos 2t,\sin 2t} \right)$
From the result in Exercise 11 we obtain the space curve in Figure 8 (B) is a helix moving above a circle in the $yz$-plane, while its height is $x=t$. And also the projection of the space curve onto the $xy$-plane is a wave. Thus, the space curve is Figure 8 (B).
(c) ${{\bf{r}}_3}\left( t \right) = \left( {1,t,t} \right)$
From the result in Exercise 11 we obtain the projection of the space curve in Figure 8 (A) onto the $xy$-plane is a vertical line. Thus, the space curve is Figure 8 (A).
