Prove that the necessary and sufficient condition for three vectors `veca,vecb` and `vecc` to be coplanar is that there exist scalars `l,m,n` (not all zero simultaneously) such that `lveca+mvecb+nvecc=vec0`.

Three vectors vecA, vecB and vecC add up to zero. Find which is false.

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.

Give a condition that the three vectors veca,vecb and vecc form the three sides of a triangle. What are other possibilities ?

If the vectors veca, vecb, vecc are non -coplanar and l,m,n are distinct scalars such that [(lveca+mvecb+nvecc, lvecb+mvecc+nveca,lvecc+mveca+nvecb)]=0 then

Given three vectors veca, vecb and vecc are non-zero and non-coplanar vectors. Then which of the following are coplanar.

If non-zero vectors veca and vecb are equally inclined to coplanar vector vecc , then vecc can be

The scalars l and m such that lveca + m vecb =vecc, " where " veca, vecb and vecc are given vectors, are equal to

For any three vectors veca,vecb,vecc show that veca-vecb,vecb-vecc,vecc-veca are coplanar.

Statement-I (S_1) : If A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) are non-collinear points. Then, every point (x, y) in the plane of triangleABC , can be expressed in the form ((kx_1+lx_2+mx_3)/(k+l+m), (ky_1+ly_2+my_3)/(k+l+m)) Statement-II (S_2) The condition for coplanarity of four A(a), B(b), C(c), D(d) is that there exists scalars l, m, n, p not all zeros such that la+mb+nc+pd=0 where l+m+n+p=0 .

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