The position vectors of the points `Pa n dQ` with respect to the origin `O` are ` vec a= hat i+3 hat j-2 hat k` and ` vec b=3 hat i- hat j-2 hat k ,` respectively. If `M` is a point on `P Q ,` such that `O M` is the beisector of `P O Q ,` then ` vec O M` is a. `2( hat i- hat j+ hat k)` b. `2 hat i+ hat j-2 hat k` c. `2(- hat i+ hat j- hat k)` d. `2( hat i+ hat j+ hat k)`

Options

A. `2(hati-hatj+hatk)`
B. `2 hati+hatj-2hatk`
C. `2(-hati+hatj-hatk)`
D. `2(hati+hatj+hatk)`

Answer

B

Explanation

Since `|vec(OP)| = |vec(OQ)| = sqrt(14), Delta OPQ` is isosceles.
Hence, the internal bisector OM is perpendicular to PQ and M is the midpoint of P and Q. Therefore,
`vec(OM) = (1)/(2) (vec(OP)+ vec(OQ))= 2hati+hatj-2hatk`