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.

Let vec a , vec b , vec c , vec d be the position vectors of the four distinct points A , B , C , Ddot If vec b- vec a= vec c- vec d , then show that A B C D is parallelogram.

Let vec a , vec b , vec c , vec d be the position vectors of the four distinct points A , B , C , Ddot If vec b- vec a= vec c- vec a , then show that A B C D is parallelogram.

Let vec a , vec b , vec c , vec d be the position vectors of the four distinct points A , B , C , Ddot If vec b- vec a= vec a- vec d , then show that A B C D is parallelogram.

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

Statement -1 : If a transversal cuts the sides OL, OM and diagonal ON of a parallelogram at A, B, C respectively, then (OL)/(OA) + (OM)/(OB) =(ON)/(OC) Statement -2 : Three points with position vectors veca , vec b , vec c are collinear iff there exist scalars x, y, z not all zero such that x vec a + y vec b +z vec c = vec 0, " where " x +y + z=0.

Statement -1 : If veca and vecb are non- collinear vectors, then points having position vectors x_(1) vec(a) + y_(1) vec(b) , x_(2)vec(a)+ y_(2) vec(b) and x_(3) veca + y_(3) vecb are collinear if |(x_(1),x_(2),x_(3)),(y_(1),y_(2),y_(3)),(1,1,1)|=0 Statement -2: Three points with position vectors veca, vecb , vec c are collinear iff there exist scalars x, y, z not all zero such that x vec a + y vec b + z vec c = vec 0, " where " x+y+z=0.

If vec a ,\ vec b ,\ vec c are position vectors of the vertices A ,\ B\ a n d\ C respectively, of a triangle A B C ,\ write the value of vec A B+ vec B C+ vec C Adot

If vec a ,\ vec b ,\ vec c are position vectors o the point A ,\ B ,\ a n d\ C respectively, write the value of vec A B+ vec B C+ vec A Cdot

vec a , vec b and vec c are the position vectors of points A ,B and C respectively, prove that : vec a× vec b+ vec b× vec c+ vec c× vec a is vector perpendicular to the plane of triangle A B Cdot

Let vec a , vec b , vec c be unit vectors such that vec adot vec b= vec adot vec c=0 and the angle between vec ba n d vec c is pi/6,t h a t vec a=+-2( vec bxx vec c)dot