Vector Algebra

Let two non-collinear vectors \(\hat{a}\) and \(\hat{b}\) form an acute angle. A point \(\mathrm{P}\) moves, so that at any time \(t\) the position vector \(\overline{\mathrm{OP}}\), where \(\mathrm{O}\) is origin, is given by \(\hat{a} \sin t+\hat{b} \cos t\), when \(P\) is farthest from origin \(O\), let \(M\) be the length of \(\overline{\mathrm{OP}}\) and \(\hat{\mathrm{u}}\) be the unit vector along \(\overline{\mathrm{OP}}\), then