What is the value of `[vec a vec b vec c]` if `vec a, vec b, vec c` are non-zero coplanar vectors ?

If vec r* vec a =0 = vec r* vec b ,where vec a and vec b are non-coplanar vectors then

Prove that [vec a -vec b, vec b - vec c, vec c- vec a] =0

Solve the vector equation vec r xx vec a + k vec r = vec b , where vec a, vec b are two non-collinear vectors and k is any scalar.

If vec a , vec b and vec c are non-coplanar (independent) vectors, prove that the vectors vec a- 2 vec b+ 3 vec c , -2 vec a + 3 vec b- 4 vec c and vec a -vec b+ 2 vec c are also linearly independent.

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

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

For any three vectors vec a,vec b and vec c , show that vec a- vec b,vec b- vec c and vec c- vec a are coplanar.

If three unit vectors vec a , vec b , and vec c satisfy vec a+ vec b+ vec c=0, then find the angle between vec a and vec bdot

Show that vectors vec a ,\ vec b ,\ vec c are coplanar if vec a+ vec b ,\ vec b+ vec c ,\ vec c+ vec a are coplanar.

If vec r* vec a =0 = vec r* vec b =0 and also vec r* vec c =0 for some non-zero vector vec r , then the value of vec a*(vec b xx vec c) is........