`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 non - coplanar vectors vec (r) . vec(a) = vecr . vec(b) = vecr.vec(c ) = 0 , show that vec (r ) is a zero vector .

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

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

Let vec(a),vec(b)andvec(c) be three non-coplanar vectors and let vec(p),vec(q),vec(r) be the vectors defined by the relations 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)])" Then the value of "(vec(a)+vec(b))*vec(p)+(vec(b)+vec(c))*vec(q)+(vec(c)+vec(a))*vec(r)=