Prove that a necessary and sufficient condition for three vectors ` vec a , vec b` and ` vec c` to be coplanar is that there exist scalars `l , m , n` not all zero simultaneously such that `l vec a+m vec b+n vec c= vec0dot`

Three points with position vectors vec(a), vec(b), vec(c ) will be collinear if there exist scalars x, y, z such that

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

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 a , vec ba n d vec c are unit coplanar vectors, then the scalar triple product [2 vec a- vec b2 vec b- vec c2 vec c- vec a] is 0 b. 1 c. -sqrt(3) d. sqrt(3)

If vec a ,\ vec b ,\ vec c are three vectors such that vec adot vec b= vec adot vec c then show that vec a=0\ or ,\ vec b=c\ or\ vec a_|_( vec b- vec c)dot

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

Let vec a , vec b , vec c are three non-zero vectors such that vec axx vec b= vec ca n d vec bxx vec c= vec a ; prove that vec a , vec b , vec c are mutually at righ angles such that | vec b|=1a n d| vec c|=| vec a|dot

Let vec a , vec b , vec c be the position vectors of three distinct points A, B, C. If there exist scalars x, y, z (not all zero) such that x vec a+y vec b+z vec c=0a n dx+y+z=0, then show that A ,Ba n dC lie on a line.

Prove that ( vec a- vec b)dot{( vec b- vec c)xx( vec c- vec a)}=0

The position vectors of the vertices A ,Ba n dC of a triangle are three unit vectors vec a , vec b ,a n d vec c , respectively. A vector vec d is such that vecd dot vec a= vecd dot vec b= vec d dot vec ca n d vec d=lambda( vec b+ vec c)dot Then triangle A B C is a. acute angled b. obtuse angled c. right angled d. none of these