Summary: Suppose that \(p\) and \(q\) are projections in a unital \(C^*\)-algebra \(\mathfrak{A}\) such that \(\Vert p(1-q)\Vert <1\). It is shown that there exists a unitary \(u\) in \(\mathfrak{A}\) which is homotopic to the unit of \(\mathfrak{A} \), and satisfies \(pup=pu^*p\), \(u(pqp)u^*=qpq\) and \[ \Vert 1-u\Vert \le \sqrt{\frac{2\Vert (qp)^† \Vert }{1+\Vert (qp)^† \Vert }}\cdot \Vert p(1-q)\Vert, \] where \((qp)^†\) denotes the Moore-Penrose inverse of \(qp\). Under the same restriction of \(\Vert p(1-q)\Vert <1\), it is proved that \(\Vert p-q\Vert <1\) if and only if there exists a unitary \(u\) in \(\mathfrak{A}\) such that \(pup\) is normal and \(q=upu^*\). An example is constructed to show that there exist certain Hilbert space \(H\) and projections \(p\) and \(q\) on \(H\) such that \(\Vert p-q\Vert =1\) and \(q=upu^*\) for some unitary operator \(u\) on \(H\).