If ` vec r_1, vec r_2, vec r_3` are the position vectors of the collinear points and scalar `p a n d q` exist such that ` vec r_3=p vec r_1+q vec r_2,` then show that `p+q=1.`

Let OP,OQ, OR are three edges of a regular tetrahedron of edge length a . If vec p , vec q and vec r are the position vectors of the points P,Q and R & O is the origin then |vec p timesvec q+vec q timesvec r+vec r timesvec p| is equal to

P(vec p) and Q(vec q) are the position vectors of two fixed points and R(vec r) is the position vectorvariable point.If R moves such that (vec r-vec p)xx(vec r-vec q)=0 then the locus of R is

Let vec r_1, vec r_2, vec r_3, , vec r_n be the position vectors of points P_1,P_2, P_3 ,P_n relative to the origin Odot If the vector equation a_1 vec r_1+a_2 vec r_2++a_n vec r_n=0 hold, then a similar equation will also hold w.r.t. to any other origin provided a. a_1+a_2+dot+a_n=n b. a_1+a_2+dot+a_n=1 c. a_1+a_2+dot+a_n=0 d. a_1=a_2=a_3dot+a_n=0

If vec PO+vec OQ=vec QO+vec OR, show that the point,P,Q,R are collinear.

Statement 1: Let vec a , vec b , vec ca n d vec d be the position vectors of four points A ,B ,Ca n dD and 3 vec a-2 vec b+5 vec c-6 vec d=0. Then points A ,B ,C ,a n dD are coplanar. Statement 2: Three non-zero, linearly dependent coinitial vector ( vec P Q , vec P Ra n d vec P S) are coplanar. Then vec P Q=lambda vec P R+mu vec P S ,w h e r elambdaa n dmu are scalars.

If vec r=3vec p+4vec q and 2vec r=vec p-3vec q then

if vec(P) xx vec(R ) = vec(Q) xx vec(R ) , then

If vec p xxvec q=vec r and vec p*vec q=c, then vec q is

If P,Q and R are three collinear points such that vec PQ=vec a and vec QR=vec b. Find the vector vec PR.

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