Let two non-collinear unit vectors ^a and ^b form an acute angle. A point P moves so that at any time t the position vector
→OP (where, O is the origin) is given by
^acos t+^bsin t. When P is farthest from origin O, let M be the length of →OP and ^u be the unit vector along →OP. Then,
^u=^a+^b|^a+^b| and M=(1+^a.^b)12
Given, →OP=^a cos t+^b sin t⇒|→OP|=√(^a.^a) cos2 t+^b.^b sin2 t+2^a.^b sin t cos t⇒|→OP|=√1+^a.^b sin 2t⇒|→OP|max=M=√1+^a.^b at sin 2t=1⇒t=π4At t=π4, →OP=1√2(^a+^b)
Unit vector along →OP at (t=π4)=^a+^b|^a+^b|