In the Cartesian plane, a man starts at origin and walks a distance of 3 units of the north-east direction and reaches a point P. from P, he walks a distance of 4 units in the north-west direction to reach a point Q. construct the parallelogram OPQR with OP and PQ as adjacent sides. let M be the mid-point of PQ.

Answer

A, B, C, D

Explanation

(A) Let `hati and hatj` be the unit vectors along OX and OY respectively.
Now, OP=3 and `anngleXOP=45^(@)` implies that
`OP=(3cos45^(@))hati+(3sin45^(@))hatj=(3)/(sqrt(2))(hati+hatj)`
(B) Again `angleXOR=135^(@) and OR=4` implies that
`OR=(4)/(sqrt(2))(-hati+hatj)=2sqrt(2)(-hati+hatj)`
(C) the position vector of Q is given by
`OP+PQ=OP+OR=(1)/(sqrt(2))(-hati+7hatj)`

`thereforeOM=(((3)/(sqrt(2)))(hati+hatj)+((1)/(sqrt(2)))(-hati+7hatj))/(2)`
`=(2hati+10hatj)/(2sqrt(2))=(hati+5hatj)/(sqrt(2))`
(D) Now, `PT:RT=1:2`
thereofore, `OT=(1(OR)+2(OP))/(3)`
`=(((4)/(sqrt(2)))(-hati+hatj)+2[((3)/(sqrt(2)))(hati+hatj)])/(3)`
`=(sqrt(2))/(3)(hati+5hatj)`