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 , 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 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 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]) .

If vec a and vec b are two vectors, then prove that ( vec axx vec b)^2=|[vec a.vec a,vec a.vec b],[vec b.vec a,vec b.vec b]| .

Let vec a , vec b and vec c be three non-zero vectors such that vec a+ vec b+ vec c=0 and lambda vec bxx vec a+ vec bxx vec c+ vec cxx vec a=0, then find the value of lambda

If vec a , vec b ,a n d vec c are three non-coplanar vectors, then find the value of ( vec a .( vec bxx vec c))/( vec b .( vec cxx vec a))+( vec b .( vec cxx vec a))/( vec c .( vec axx vec b))+( vec c . ( vec bxx vec a))/( vec a . ( vec bxx vec c))

If vec a , vec b and vec c are three non-coplanar vectors, then ( vec a+ vec b+ vec c).[( vec a+ vec b)xx( vec a+ vec c)] equals a. 0 b. [ vec a vec b vec c] c. 2[ vec a vec b vec c] d. -[ 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,