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 be the position vectors of three distinct points A,B,C. If there exist scalars x, y,Z (not all zero) such that xvec a+yvec b+zvec c=0 and x+y+z=0, then show that A,B and C 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,D. If vec b-vec a=vec a-vec d, then show that ABCD 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.

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

If vec a,vec b,vec c are the position vectors of points A,B,C and D respectively such that (vec a-vec d)*(vec b-vec c)=(vec b-vec d)*(vec c-vec a)=0 then D is the

veca , vec b , vec c are non-coplanar vectors and x vec a + y vec b + z vec c = vec 0 then

Let vec a, vec b, vec c be three non-zero vectors such that [vec with bvec c] = | vec a || vec b || vec c | then

Let vec a , vec b , vec c be the three unit vectors such that vec a+5 vec b+3 vec c= vec0 , then vec a. ( vec bxx vec c) is equal to