= Solution
{c}
Let $v\in S$. Choose a <unit vector> $u$ perpendicular to $v$, and put $w=v\times u$. The <cross product> identities show that $(v,u,w)$ is an <orthonormal basis> with positive orientation: $u\times(v\times u)=v$, so
$$
\det\begin{pmatrix}v&u&w\end{pmatrix}
=v\cdot(u\times w)=v\cdot v=1.
$$
Thus the <matrix> $A_v$ with these columns belongs to the <special orthogonal group> $G$ and satisfies $A_vN=v$. For any $v,z\in S$, the element $A_zA_v^{-1}$ sends $v$ to $z$. Therefore the <group action> on the <unit sphere> is \b[transitive].
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