Let ` vec a , vec b ,a n d vec ca n d vec a^' , vec b^' , vec c '` are reciprocal system of vectors, then prove that ` vec a^'xx vec b^'+ vec b^'xx vec c^'+ vec c^'xx vec a^'=( vec a+ vec b+ vec c)/([ vec a vec b vec c])` .

Let vec a , vec b ,and vec c be any three vectors, then prove that [ vec axx vec b vec bxx vec c vec cxx vec a ]= [vec a vec b vec c]^2

Prove that [[ vec a+ vec b, vec b+ vec c, vec c+ vec a]]=2[ [vec a, vec b, vec c]]dot

If vec a+ vec b+ vec c = vec 0 then prove that vec axx vec b= vec bxx vec c = vec cxxvec a .

Let vec a , vec b ,a n d vec c be non-coplanar vectors and let the equation vec a^' , vec b^' , vec c ' are reciprocal system of vector vec a , vec b , vec c , then prove that vec axx vec a^'+ vec bxx vec b^'+ vec cxx vec c ' is a null vector.

If vec a , vec b , and vec c are three vectors such that vec axx vec b= vec c , vec bxx vec c= vec a , vec cxx vec a= vec b , then prove that | vec a|=| vec b|=| vec c| .

If vec a , vec b , vec ca n d vec d are distinct vectors such that vec axx vec c= vec bxx vec da n d vec axx vec b= vec cxx vec d , prove that ( vec a- vec d). (vec b- vec c)!=0,

If vec a , vec b , vec ca n d vec d are distinct vectors such that vec axx vec c= vec bxx vec da n d vec axx vec b= vec cxx vec d , prove that ( vec a- vec d). (vec b- vec c)!=0,

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 vec a , vec b , vec ca n d vec d are the position vectors of the vertices of a cyclic quadrilateral A B C D , prove that (| vec axx vec b+ vec bxx vec d+ vec d xx vec a|)/((vec b- vec a).(vec d- vec a)) + (| vec bxx vec c+ vec cxx vec d+ vec d xx vec b|)/((vec b- vec c).( vec d- vec c))=0dot

Prove that , vec(a) xx ( vec (b) + vec(c)) + vec(b) xx (vec(c) + vec(a)) + vec(c)xx(vec (a) + vec(b)) = vec(0)