The position vectors of the points P,Q,R,S are `hati++hatj+hatk, 2hati+5hatj,3hati+2hatj and hati-6hatj-hatk.` Prove that the lines PQ and RS are pralel and find the ratio tof their lengths.

The position vectors of the point P,Q,R and S are respectively hati+hatj+hatk, 2hati+5hatj, 3hati+2hatj-3hatk and hati-6hatj-hatk . Prove that the lines PQ and RS are parallel and the ratio of their length is 1/2

If the position vectors of the vertices of a triangle be 2hati+4hatj-hatk,4hati+5hatj+hatk and 3 hati+6hatj-3hatk , then the triangle is

If the position vectors of P and Q are (hati+3hatj-7hatk) and (5hati-2hatj+4hatk) , then |PQ| is

If the position vectors of the three points A,B,C are 2hati+4hatj-hatk, hati+2hatj-3hatk and 3hati+hatj+2hatk respectively, find a vector perpendicular to the plane ABC.

If the figure formed by the four points hati+hatj-hatk,2hati+3hatj,3hati+5hatj-2hatk and hatk-hatj is

The position vectors of the point A, B, C and D are 3hati-2hatj -hatk, 2hati+3hatj-4hatk, -hati+hatj+2hatk and 4hati+5hatj +lamda hatk , respectively. If the points A, B, C and D lie on a plane, find the value of lamda .

The points with position vectors alpha hati+hatj+hatk, hati-hatj-hatk, hati+2hatj-hatk, hati+hatj+betahatk are coplanar if