Given two distinct nonzero vectors $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ in 3 dimensions, define a sequence of vectors by
$$\mathbf { v } _ { n + 2 } = \mathbf { v } _ { n } \times \mathbf { v } _ { n + 1 } \left( \text { so } \mathbf { v } _ { 3 } = \mathbf { v } _ { 1 } \times \mathbf { v } _ { 2 } , \mathbf { v } _ { 4 } = \mathbf { v } _ { 2 } \times \mathbf { v } _ { 3 } \text { and so on } \right) .$$
Let $S = \left\{ \mathbf { v } _ { n } \mid n = 1,2 , \ldots \right\}$ and $U = \left\{ \left. \frac { \mathbf { v } _ { n } } { \left| \mathbf { v } _ { n } \right| } \right\rvert\, n = 1,2 , \ldots \right\}$. (Note: Here $\times$ denotes the cross product of vectors and $| \mathbf { v } |$ denotes the magnitude of the vector $\mathbf { v }$. The vector $\mathbf { 0 }$ with 0 magnitude, if it occurs in $S$, is counted. But in that case of course the $\mathbf { 0 }$ vector is not considered while listing elements of $U$.)
(a) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 2.
(b) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 3.
(c) There exist vectors $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$ for which the cardinality of $S$ is 4.
(d) Suppose that for some $\mathbf { v } _ { \mathbf { 1 } }$ and $\mathbf { v } _ { \mathbf { 2 } }$, the set $S$ is infinite. Then the set $U$ is also infinite.