Point `P` divides line joining A`(hati+hatj+hatk)` and `B(2hati+hatj+3hatk)` in the ratio `lambda:1` such that `(rarrOB.raarOP-3|rarrOAxxrarrOP|^2)=6. Find `lambda`.

Find the position vector of a point R which divides the line joining the point P(hati + 2hatj - hatk) and Q(-hati + hatj + hatk) in the ratio 2 : 1 , (i) internally and (ii) externally.

Find the position vector of a point R which divides the line joining two points P(hati+2hatj-hatk) and Q(-hati-hatj+hatk) in the ratio 2:1: internally

Find the position vector of a point R which divides the line joining two points P(hati-2hatj-hatk) and Q(-hati-hatj+hatk) in the ratio 2:1: externally

The lines barr=hati+2hatj+3hatk+lambda(hati+2hatj+3hatk)andbarr=-2hatj+hatk+lambda(2hati+2hatj-2hatk) are

Find the angle between the line vecr = (hati +2hatj -hatk ) +lambda (hati - hatj +hatk) and vecr cdot (2hati - hatj +hatk) = 4.

Find the mid point of the ine segment joining the points P(2hati+3hatj+3hatk) and Q(4hati+hatj-2hatk)

Find the angle between the line barr = (hati + 2hatj + hatk) + lambda(hati +hatj + hatk) and the plane barr*(2hati - hatj + hatk) = 5 .

Find the angle between the line vecr=(hati+hatj-2hatk)+lambda(hati-hatj+hatk) and the plane vecr.(2hati-hatj+hatk)=4 .