If ` vec a , vec ba n d vec c` are three non coplanar vectors, then prove that ` vec d=( vec adot vec d)/([ vec a vec b vec c])( vec bxx vec c)+( vec bdot vec d)/([ vec a vec b vec c])( vec cxx vec a)+( vec cdot vec d)/([ vec a vec b vec c])( vec axx vec b)`

[vec a, vec b + vec c, vec d] = [vec a, vec b, vec d] + [vec a, vec c, vec d]

Let vec a , vec ba n d vec c be three non-coplanar vectors and vec p , vec qa n d vec r the vectors defined by the relation vec p=( vec bxx vec c)/([ vec a vec b vec c]), vec q=( vec cxx vec a)/([ vec a vec b vec c])a n d vec r=( vec axx vec b)/([ vec a vec b vec c])dot Then the value of the expression ( vec a+ vec b)dot vec p+( vec b+ vec c)dot vec q+( vec c+ vec a)dot vec r is a. 0 b. 1 c. 2 d. 3

If vec a , vec b and vec c are three non-zero, non coplanar vector vec b_1= vec b-( vec bdot vec a)/(| vec a|^2) vec a , vec c_1= vec c-( vec cdot vec a)/(| vec a|^2) vec a+( vec bdot vec c)/(| vec c|^2) vec b_1 , , c_2= vec c-( vec cdot vec a)/(| vec a|^2) vec a-( vec bdot vec c)/(| vec b_1|^2) , b_1, vec c_3= vec c-( vec cdot vec a)/(| vec c|^2) vec a+( vec bdot vec c)/(| vec c|^2) vec b_1 , vec c_4= vec c-( vec cdot vec a)/(| vec c|^2) vec a=( vec bdot vec c)/(| vec b|^2) vec b_1 then the set of orthogonal vectors is ( vec a , vec b_1, vec c_3) b. ( vec a , vec b_1, vec c_2) c. ( vec a , vec b_1, vec c_1) d. ( vec a , vec b_2, vec c_2)

If vec a,vec b, and vec c are three non-coplanar vectors,then find the value of (vec a*(vec b xxvec c))/(vec b*(vec c xxvec a))+(vec b*(vec c xxvec a))/(vec c*(vec a xxvec b))+(vec c*(vec b xxvec a))/(vec a xxvec c))

[vec a + vec b, vec b + vec c, vec c + vec a] = 2 [vec a, vec b, vec c]

The length of the perpendicular form the origin to the plane passing through the point a and containing the line vec r= vec b+lambda vec c is a. ([ vec a vec b vec c])/(| vec axx vec b+ vec bxx vec c+ vec cxx vec a|) b. ([ vec a vec b vec c])/(| vec axx vec b+ vec bxx vec c|) c. ([ vec a vec b vec c])/(| vec bxx vec c+ vec cxx vec a|) d. ([ vec a vec b vec c])/(| vec cxx vec a+ vec axx vec b|)

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|=odot

If vec a, vec b , vec c are three non- coplanar vectors such that vec a + vec b + vec c = alpha vec d and vec b +vec c + vec d = beta vec a, " then " vec a + vec b + vec c + vec d to equal to

If vec a,vec b and vec c are three non-zero vectors,prove that [vec a+vec b,vec b+vec c,vec c+vec a]=2[vec a,vec b,vec c]

If vec a , vec b , vec c are three non coplanar vectors such that vec ddot vec a= vec ddot vec b= vec ddot vec c=0, then show that d is the null vector.