If ` vec a , vec b ,a n d 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|dot`

If vec a, vec b, vec c are three given vectors show that [ vec a + vecb + vec c , vecb + vec c , vec a + vec b + vec c ]=0.

Let vec a , vec b ,a n d 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]^2dot

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 c are unit vectors such that vec a+ vec b+ vec c =0 , find the value of vec a.vec b+ vec b .vec c + vec c. vec a .

If vec a and vec b are two vectors such that | vec axx vec b|=2, then find the value of [ vec a vec b vec axx vec b].

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

Prove that [ vec a, vec b, vec c + vec d] = [ vec a, vec b, vec c] + [ vec a, vec b , vec d] .

If vec a , vec b ,a n d vec c are three non-coplanar non-zero vecrtors, then prove that ( vec a . vec a) vec bxx vec c+( vec a . vec b) vec cxx vec a+( vec a . vec c) vec axx vec b=[ vec b vec c vec a] vec a

For any three vectors vec a, vec b , vec c , show that vec a xx (vec b + vec c) + vec b xx (vec c + vec a) + vec c xx (vec a + vec b) = 0

If vectors vec a , vec b ,a n d vec c are coplanar, show that | vec a vec b vec c vec adot vec a vec adot vec b vec adot vec c vec bdot vec a vec bdot vec b vec bdot vec c|=0