If a, b, c and a', b', c' are recoprocal system of vectors, then prove that `a'timesb'+b'timesc'+c'timesa'=(a+b+c)/([a b c])`.

Let vec a,vec b, and 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 a xxvec a'+vec b xxvec b'+vec c xxvec c ,is a null vector.

If a, b, c are vectors such that [a b c] = 4 then [a times b" " btimesc" "c timesa]=

If a, b and c are three non-coplanar vectors, then find the value of (a*(btimesc))/(ctimes(a*b))+(b*(ctimesa))/(c*(atimesb)) .

If a, b and c are three vectors such that [a b c]=1, then find the value of [a+b b+c c+a]+[ atimesb btimesc ctimesa ]+[ atimes(btimesc) btimes(ctimesa) ctimes(atimesb) ]

If a,b and c are any three vectors in space, then show that (c+b)xx(c+a)*(c+b+a)= [a b c]

If a+b+c!=0 and |a b c b c a c a b|=0 , then prove that a=b=cdot

If a,b and c are non-coplanar vectors, prove that 3a-7b-4c, 3a-2b+c and a+b+2c are complanar.