Value of `[vec a xx vec b,vec a xx vecc,vec d]` is always equal to

Let vec a , vec ba n d vec c be three non-coplanar vecrors and vec r be any arbitrary vector. Then ( vec axx vec b)xx( vec rxx vec c)+( vec bxx vec c)xx( vec rxx vec a)+( vec cxx vec a)xx( vec rxx vec b) is always equal to [ vec a vec b vec c] vec r b. 2[ vec a vec b vec c] vec r c. 3[ vec a vec b vec c] vec r d. none of these

Let vec a , vec ba n d vec c be three non-coplanar vecrors and vec r be any arbitrary vector. Then ( vec axx vec b)xx( vec rxx vec c)+( vec bxx vec c)xx( vec rxx vec a)+( vec cxx vec a)xx( vec rxx vec b) is always equal to [ vec a vec b vec c] vec r b. 2[ vec a vec b vec c] vec r c. 3[ vec a vec b vec c] vec r d. none of these

Let vec a , vec ba n d vec c be three non-coplanar vecrors and vec r be any arbitrary vector. Then ( vec axx vec b)xx( vec rxx vec c)+( vec bxx vec c)xx( vec rxx vec a)+( vec cxx vec a)xx( vec rxx vec b) is always equal to [ vec a vec b vec c] vec r b. 2[ vec a vec b vec c] vec r c. 3[ vec a vec b vec c] vec r d. none of these

vec b and vec c are unit vectors.Then for any arbitrary vector vec a,(((vec a xxvec b)+(vec a xxvec c))xx(vec b xxvec c))vec b-vec c is always equal to a.|vec a| b.(1)/(2)|vec a| c.(1)/(3)|vec a| d.none of these

If vec a = vec b + vec c, vec b xxvec d = vec 0, vec c * vec d = 0 then (vec d xx (vec a xxvec d)) / (| vec d | ^ (2)) is always equal to

The triple product of (vec d+vec a)*[vec a xx(vec b xx(vec c xxvec d))] is equal to:

For any three vectors vec a, vec b, vec c, (vec a-vec b) * (vec b-vec c) xx (vec c-vec a) is equal to

If vec a, vec b and vec c are non-coplanar vector and vec a times vec c is perpendicular to vec a times(vec b times vec c) ,then the value of [vec a times(vec b times vec c)]times vec c is equal to :

If vec a,vec c,vec d are non-coplanar vectors,then vec d*{vec a xx[vec b xx(vec c xxvec d)]} is equal to

[( vec axx vec b)xx( vec bxx vec c)( vec bxx vec c)xx( vec cxx vec a)( vec cxx vec a)xx( vec axx vec b)] is equal to (where vec a , vec ba n d vec c are nonzero non-coplanar vector) [ vec a vec b vec c]^2 b. [ vec a vec b vec c]^3 c. [ vec a vec b vec c]^4 d. [ vec a vec b vec c]