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 a t+ b t . When P is farthest from origin O , let M be the length of OP and u be the unit vector along OP , then

Step-by-step solution

aligned M & =| O P | M & = a t+ b t & = ( a t)^2+( b t)^2+2( a t) ( b t) & = ^2 t+ ^2 t+ a b (2 t t) & = 1+ a b ( 2 t) aligned Maximum value of 2 t=1 array ll & 2 t= ⁻¹(1) & t= 4 array M= 1+ a b (1) =(1+ a b )^ 1 2 Now, u = OP | OP | = a t+ b t | a t+ b t| = a ( 1 2 )+ b ( 1 2 ) | a ( 1 2 )+ b ( 1 2 ) | Unit vector of OP is u = a + b |( a + b )|