If 2` vec A C` = 3` vec C B` , then prove that 2` vec O A` =3` vec C B` then prove that 2` vec O A` + 3` vec O B` =5` vec O C` where `O` is the origin.

if vec Ao + vec O B = vec B O + vec O C ,than prove that B is the midpoint of AC.

If vec a and vec b are orthogonal vectors, prove that (vec a + vec b)^(2) = (vec a - vec b)^(2) .

If A B C D is a rhombus whose diagonals cut at the origin O , then proved that vec O A+ vec O B+ vec O C+ vec O D+ vec Odot

If A B C D is a rhombus whose diagonals cut at the origin O , then proved that vec O A+ vec O B+ vec O C+ vec O D+ vec Odot

Statement 1:Let A( vec a),B( vec b)a n dC( vec c) be three points such that vec a=2 hat i+ hat k , vec b=3 hat i- hat j+3 hat ka n d vec c=- hat i+7 hat j-5 hat kdot Then O A B C is a tetrahedron. Statement 2: Let A( vec a),B( vec b)a n dC( vec c) be three points such that vectors vec a , vec ba n d vec c are non-coplanar. Then O A B C is a tetrahedron where O is the origin.

Prove that [ vec a, vec b, vec c + vec d] = [ vec a, vec b, vec c] + [ vec a, vec b , vec d] .

For any three vectors veca, vec b, vec c prove that (vec a + vec b)+ vec c = vec a + (vec b + vec c)

Prove that vec a xx (vec b + vec c) + vec b xx(vec a + vec c)+ vec c xx(vec a + vec b) = vec 0

If vec ba n d vec c are two-noncollinear vectors such that vec a||( vec bxx vec c), then prove that ( vec axx vec b) . ( vec axx vec c) is equal to | vec a|^2( vec bdot vec c)dot

A , B , C , D are any four points, prove that vec A B dot vec C D+ vec B Cdot vec A D+ vec C Adot vec B D=0.