`vec(a),vec(b),vec(c)` are the three coplanar vectors and if `vec(r).vec(a)=vec(r).vec(b)=vec(r).vec(c)=0` then prove that ` vec(r)` is a zero vector

If vec a,vec b,vec c are three non coplanar vectors such that vec a.vec a*vec a=vec dvec b=vec d*vec c=0 then show that vec d is the null vector.

If vec a , vec b , vec c are three non coplanar vectors such that vec ddot vec a= vec ddot vec b= vec ddot vec c=0, then show that d is the null vector.

If vec(a), vec(b), vec(c ) are three vectors such that vec(a)+vec(b)+vec(c )=0 , then prove that : vec(a)xx vec(b)=vec(b)xx vec(c )=vec(c )xx vec(a) and hence, show that [vec(a)vec(b)vec(c )]=0 .

Let vec(a), vec(b), vec(c) be non-coplanar vectors and vec(p)=(vec(b)xxvec(c))/([vec(a)vec(b)vec(c)]), vec(q)=(vec(c)xxvec(a))/([vec(a)vec(b)vec(c)]), vec(r)=(vec(a)xxvec(b))/([vec(a)vec(b)vec(c)]) . What is the value of (vec(a)-vec(b)-vec(c)).vec(p)+(vec(b)-vec(c)-vec(a)).vec(q)+(vec(c)-vec(a)-vec(b)).vec(r) ?

For any three coplanar vectors vec(a), vec(b), vec(c ) , show that vec(a)-vec(b), vec(b)-vec(c ), vec(c )-vec(a) are coplanar.

Let vec a , vec b ,vec c are three non- coplanar vectors such that vecr_(1)=veca + vec c , vecr_(2)= vec b+vec c -veca , vec r_(3) = vec c + vec a + vecb, vec r = 2 vec a - 3 vec b + 4 vec c. If vec r = lambda_(1)vecr_(1)+lambda_(2)vecr_(2)+lambda_(3)vecr_(3) , then

Let vec a xx vec b,vec b xx vec c,vec c xx vec a are non coplanar vectors and [[vec a,vec b,vec c]]=1.vec r is a vector such that vec r.vec a=vec r.vec b=vec r.vec c=2, then area of triangle whose vertices are vec a,vec b and vec c is (A) |vecr|/2 (2) 2|vecr| (C) |vecr|/4 (D) 4|vecr|

If vec a, vec b, vec c are three non-coplanar vectors and vec p, vec q, vec r are vectors defined by the relations vec p = (vec b xxvec c) / ([vec avec bvec c]), vec q = (vec c xxvec a) / ([vec avec bvec c]), vec r = (vec a xxvec b) / ([vec avec bvec c]) then the value of expression (vec a + vec b) * vec p + (vec b + vec c) * vec q + (vec c + vec a) * vec r is equal to