The vertices `A , B , C` of triangle `A B C` have respectively position vectors ` vec a , vec b , vec c` with respect to a given origin `O` . Show that the point `D` where the bisector of `/_A` meets `B C` has position vector ` vec d=(beta vec b+gamma vec c)/(beta+gamma),` where `beta=| vec c- vec a|` and, `gamma=| vec a- vec b|dot`

Show that the vectors vec a - vec b, vec b - vec c , vec c - vec a are coplaner.

If vec a ,\ vec b ,\ vec c are position vectors of the vertices of a triangle, then write the position vector of its centroid.

If O A B C is a tetrahedron where O is the orogin anf A ,B ,a n dC are the other three vertices with position vectors, vec a , vec b ,a n d vec c respectively, then prove that the centre of the sphere circumscribing the tetrahedron is given by position vector (a^2( vec bxx vec c)+b^2( vec cxx vec a)+c^2( vec axx vec b))/(2[ vec a vec b vec c]) .

Given that vec adot vec b= vec adot vec c , vec axx vec b= vec axx vec c and vec a is not a zero vector. Show that vec b= vec c dot

Show that the point A ,B ,C with position vectors vec a-2 vec b+3 vec c ,2 vec a+3 vec b-4 vec c and -7 vec b+10 vec c are collinear.

Show that the point A ,B ,C with position vectors vec a-2 vec b+3 vec c ,2 vec a+3 vec b-4 vec c and -7 vec b+10 vec c are collinear.

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.

If A,B and C are the vertices of a triangle with position vectors vec(a) , vec(b) and vec(c ) respectively and G is the centroid of Delta ABC , then vec( GA) + vec( GB ) + vec( GC ) is equal to

For any three vectors a ,\ b ,\ c prove that [ vec a+ vec b\ vec b+ vec c\ vec c+ vec a]=2\ [ vec a\ vec b\ vec c]dot

Show that the points A ,B ,C with position vectors -2 vec a+3 vec b+5 vec c , vec a+2 vec b+3 vec c a n d7 vec a- vec c respectively, are collinear.