Given a vector \(U=(u_1,u_2,u_3)^\top\in R^3\) of length one and angle of rotation \(r\), a \(3\times 3\) rotation
matrix is formed using Rodrigues' formula
$$\cos(r)I_{3\times 3}+\sin(r)\Phi(U)+(1-\cos(r))UU^\top$$
where \(I_{3\times 3}\) is the \(3\times 3\) identity matrix, \(\Phi(U)\) is a \(3\times 3\) skew-symmetric matrix
with upper triangular elements \(-u_3\), \(u_2\) and \(-u_1\) in that order.
For the same vector and angle a quaternion is formed according to $$q=[cos(\theta/2),sin(\theta/2)U]^\top.$$