OX, OY and OZ are three edges of a cube andn P,Q,R are the vertices of rectangle OXPY, OXQZ and OYSZ respectively. If `vec(OX)=vecalpha, vec(OY)=vecbeta and vec(OZ)=vecgamma express vec(OP), vec(OQ), vec(OR) and vec(OS) in erms of vecalpha, vecbeta and vecgamma.

The unit vector bisecting vec(OY) and vec(OZ) is

Consider the cube in the first octant with sides OP,OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where O(0,0,0) is the origin. Let S(1/2,1/2,1/2) be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If vec(p)=vec(SP), vec(q)=vec(SQ), vec(r)=vec(SR) and vec(t)=vec(ST) then the value of |(vec(p)xxvec(q))xx(vec(r)xx(vec(t))| is

Three vectors vec(P) , vec(Q) and vec( R) are such that |vec(P)| , |vec(Q )|, |vec(R )| = sqrt(2) |vec(P)| and vec(P) + vec(Q) + vec(R ) = 0 . The angle between vec(P) and vec(Q) , vec(Q) and vec(R ) and vec(P) and vec(R ) are

Suppose that vec p, vec q and vec r are three non-coplanar vectors in R^3. Let the components of a vector vec s along vec p, vec q and vec r be 4, 3 and 5, respectively. If the components of this vector vec s along (-vec p+vec q +vec r), (vec p-vec q+vec r) and (-vec p-vec q+vec r) are x, y and z, respectively, then the value of 2vec x+vec y+vec z is

For three vectors vec p,vec q and vec r if vec r=3vec p+4vec q and 2vec r=vec p-3vec q then

Suppose that vec p,vecqand vecr are three non- coplaner in R^(3) ,Let the components of a vector vecs along vecp , vec q and vecr be 4,3, and 5, respectively , if the components this vector vec s along (-vecp+vec q +vecr),(vecp-vecq+vecr) and (-vecp-vecq+vecr) are x, y and z , respectively , then the value of 2x+y+z is

Ab, AC and AD are three adjacent edges of a parallelpiped. The diagonal of the praallelepiped passing through A and direqcted away from it is vector veca . The vector of the faces containing vertices A, B , C and A, B, D are vecb and vecc , respectively , i.e. vec(AB) xx vec(AC)=vecb and vec(AD) xx vec(AB) = vecc the projection of each edge AB and AC on diagonal vector veca is |veca|/3 vector vec(AC) is

Ab, AC and AD are three adjacent edges of a parallelpiped. The diagonal of the praallelepiped passing through A and direqcted away from it is vector veca . The vector of the faces containing vertices A, B , C and A, B, D are vecb and vecc , respectively , i.e. vec(AB) xx vec(AC)=vecb and vec(AD) xx vec(AB) = vecc the projection of each edge AB and AC on diagonal vector veca is |veca|/3 vector vec(AD) is