MHT CET20239 May 2023Morning ShiftMathematicsVector AlgebraActual
Let two non-collinear 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 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
Options
- Au = a + b | a + b | and M =(1+ a b )^ 1 2
- Bu = a - b | a - b | and M =(1+ a b )^ 1 2
- Cu = a + b | a + b | and M =(1+ 2a b )^ 1 2
- Du = a - b | a - b | and M =(1- 2a b )^ 1 2
Correct answer
A. u = a + b | a + b | and M =(1+ a b )^ 1 2
Step-by-step solution
aligned & M =| OP | & M= ( a t+ b t)^2 & = ( 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 ) & Maximum value of 2 t=1 & 2 t = ⁻¹(1) & t = 4 & 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 ) | & aligned Unit vector of OP is u = a + b |( a + b )|