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(vect))| is

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(AB) is

ABCD is a regular tetrahedron P & Q are the mid -points of the edges AC and AB respectively, G is the cenroid of the face BCD and theta is the angle between the vectors vec(PG) and vec(DQ) , then

If vec a and vec b are non-collinear vectors and vectors vecalpha=(x+4y) vec a+(2x+y+1) vec b and vecbeta=(-2x+y+2) vec a+(2x-3y-1) vec b are connected by the relation 3 vecalpha=2 vecbeta ,find the value of x and y?

The position vectors of P and Q are respectively vec a and vec b . If R is a point on vec(PQ) such that vec(PR) = 5 vec(PQ), then the position vector of R, is

Let vec a , vec ba n d vec c be three non-coplanar vectors and vec p , vec qa n d vec r the vectors defined by the relation vec p=( vec bxx vec c)/([ vec a vec b vec c]), vec q=( vec cxx vec a)/([ vec a vec b vec c])a n d vec r=( vec axx vec b)/([ vec a vec b vec c])dot Then the value of the expression ( vec a+ vec b)dot vec p+( vec b+ vec c)dot vec q+( vec c+ vec a)dot vec r is a. 0 b. 1 c. 2 d. 3

Let vec a , vec ba n d vec c be three non-coplanar vectors and vec p , vec qa n d vec r the vectors defined by the relation vec p=( vec bxx vec c)/([ vec a vec b vec c]), vec q=( vec cxx vec a)/([ vec a vec b vec c])a n d vec r=( vec axx vec b)/([ vec a vec b vec c])dot Then the value of the expression ( vec a+ vec b)dot vec p+( vec b+ vec c)dot vec q+( vec c+ vec a)dot vec r is a. 0 b. 1 c. 2 d. 3

If P ,Q and R are three collinear points such that vec P Q= vec a and vec Q R = vec bdot Find the vector vec P R .