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 ,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.

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

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

If vec a,vec b,vec c are any three vectors, prove that vec a xx (vec b xx vec c) +vec b xx(vec c xx vec a)+ vec c xx(vec a xx vec b) = vec 0

Let lambda = vec a xx (vec b + vec c), vec mu = vec b xx (vec c + vec a) and vec nu = vec c xx (vec a + vec b). Then

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)dot( vec d- vec a))+(| vec bxx vec c+ vec cxx vec d+ vec d xx vec b|)/(( vec b- vec c)dot( vec d- vec c))=0dot

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)dot (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,